# TENSOR STRUCTURES ON FIBERED CATEGORIES

LUCA TERENZI

**ABSTRACT.** Let $\mathcal{S}$ be a small category admitting binary products. We show that the whole theory of monoidal $\mathcal{S}$-fibered categories, which is customarily formulated in terms of the usual internal tensor product, can be rephrased purely in terms of the associated external tensor product. More precisely, we construct a canonical dictionary relating the classical structures and properties of the internal tensor product to analogous structures and properties of the external tensor product; the translation between the internal version and the external version of the theory is mediated by suitable notions of equivalence. Our method applies to associativity, commutativity, and unit constraints, to projection formulae, as well as to monoidality of morphisms between monoidal $\mathcal{S}$-fibered categories. For instance, we show how Mac Lane's classical pentagon and hexagon axioms can be stated using the external tensor product. Our results provide a satisfactory abstract framework to study monoidal structures in the setting of perverse sheaves.

## CONTENTS

<table><tr><td>Introduction</td><td>2</td></tr><tr><td>Notation and conventions</td><td>6</td></tr><tr><td>1. Recollections on fibered categories</td><td>7</td></tr><tr><td>2. Internal and external tensor structures</td><td>14</td></tr><tr><td>3. Associativity constraints</td><td>24</td></tr><tr><td>4. Commutativity constraints</td><td>31</td></tr><tr><td>5. Unit constraints</td><td>38</td></tr><tr><td>6. Compatibility conditions</td><td>45</td></tr><tr><td>7. Projection formulae</td><td>53</td></tr><tr><td>8. Tensor structures on morphisms of fibered categories</td><td>55</td></tr><tr><td>9. Remarks on the triangulated case</td><td>71</td></tr><tr><td>References</td><td>78</td></tr></table>

LUCA TERENZI

UMPA, ENS DE LYON

46 ALLÉE D'ITALIE, 69364 LYON CEDEX 07 (FRANCE)

E-mail address: luca.terenzi@ens-lyon.fr.

2020 Mathematics Subject Classification. 18D30, 18M05.

Key words and phrases. Monoidal fibered categories, external tensor product, perverse sheaves.

The author acknowledges support by the GK 1821 "Cohomological Methods in Geometry" at the University of Freiburg and by the Labex Milyon at the ENS Lyon.
INTRODUCTION

**Motivation and goal of the paper.** Given a small category $\mathcal{S}$, the theory of $\mathcal{S}$-fibered categories can be merged with the theory of monoidal categories into a theory of monoidal $\mathcal{S}$-fibered categories. The basic definitions of this theory can be found, for example, in [Ayo07a, § 2.3.1], where fibered categories are regarded as contravariant pseudo-functors with domain $\mathcal{S}$ rather than as fibrations over $\mathcal{S}$; the same perspective is adopted throughout the present paper. In a nutshell, giving a monoidal $\mathcal{S}$-fibered category amounts to giving an $\mathcal{S}$-fibered category $\mathbb{H}$ equipped with a collection of tensor product functors

$$ - \otimes - = - \otimes_S - : \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S), \quad (A, B) \rightsquigarrow A \otimes B \quad (S \in \mathcal{S}) $$

carrying an associativity constraint (and possibly also a commutativity constraint and a unit constraint) together with, for every morphism $f : T \to S$ in $\mathcal{S}$, a natural isomorphism of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ (1) \qquad m = m_f : f^* A \otimes f^* B \xrightarrow{\sim} f^*(A \otimes B) $$

witnessing the monoidality of the inverse image functor $f^* : \mathbb{H}(S) \to \mathbb{H}(T)$. These structures are required to satisfy a certain list of compatibility conditions ensuring that any two given iterates of tensor product and inverse image functors do not get identified in more than one way. If the base category $\mathcal{S}$ admits binary products, one can also define the *external tensor product* functors

$$ - \boxtimes - = - \boxtimes_{S_1, S_2} - : \mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2), \quad (A_1, A_2) \mapsto A_1 \boxtimes A_2 \quad (S_1, S_2 \in \mathcal{S}) $$

by the formula

$$ (2) \qquad A_1 \boxtimes A_2 := \mathrm{pr}_1^{12*} A_1 \otimes \mathrm{pr}_2^{12*} A_2, $$

where $\mathrm{pr}_i^{12*} : S_1 \times S_2 \to S_i$, $i = 1, 2$, denote the canonical projections. Informally, the usual tensor product - also called the *internal tensor product* - can be recovered from the external tensor product via the formula

$$ (3) \qquad A \otimes B = \Delta_S^*(A \boxtimes B), $$

where $\Delta_S : S \hookrightarrow S \times S$ denotes the diagonal embedding. However, strictly speaking, the two constructions in (2) and (3) are not mutually inverse. Moreover, it is a priori unclear whether the monoidality isomorphisms (1) and the various additional constraints (together with their coherence and mutual compatibility conditions) can be expressed purely in terms of the external tensor product. A similar issue arises when studying monoidality of morphisms between $\mathcal{S}$-fibered categories equipped with monoidal structures. The main goal of this paper is to provide a precise answer to all these questions.

Our main motivation for this abstract work comes from the theory of perverse sheaves over algebraic or analytic varieties. To fix notation, let $k$ be a subfield of the complex numbers, and consider the category $\mathrm{Var}_k$ of (quasi-projective) algebraic $k$-varieties. To every $S \in \mathrm{Var}_k$ one can attach the algebraically constructible $\mathbb{Q}$-linear derived category $D_c^b(S, \mathbb{Q})$, a tensor-triangulated category; as $S$ varies, these categories assemble into a monoidal $\mathrm{Var}_k$-fibered category. The theory of [BBD82] yields a perverse $t$-structure on $D_c^b(S, \mathbb{Q})$ whose heart is the $\mathbb{Q}$-linear abelian category of perverse sheaves on $S$, denoted $\mathrm{Perv}(S)$. As shown by Beilinson in his celebrated paper [Bei87], the triangulated category $D_c^b(S, \mathbb{Q})$ can be canonically recovered as the bounded derived category $D^b(\mathrm{Perv}(S))$. In light of Beilinson's result, it is natural to wonder how much of the functoriality of the triangulated categories $D_c^b(S, \mathbb{Q})$ can be reconstructed directly on the level of perverse shaves.

This question is not a mere formal quirk. Over the last decades, the categories $D_c^b(S, \mathbb{Q})$ have served as a model for several other refined triangulated systems of coefficient over algebraic varieties. In some cases, it has proven successful to construct the triangulated coefficient categories directly as bounded derived categories of suitable abelian categories of enhanced perverse sheaves. The most remarkable example is M. Saito's theory of *mixed Hodge modules*, developed in [Sai90], which provides a system of coefficients enhancing the classical theory of mixed Hodge structures. Saito's triangulated categories enjoy a rich functoriality which
perfectly mimics that of the underlying constructible derived categories; in particular, they assemble into a monoidal $\mathbf{Var}_k$-fibered category. Saito's approach to the tensor structure on mixed Hodge modules uses the external perspective in an essential way: it is the internal tensor product that is defined from the external one via the above-mentioned formula (3), not the other way around. The fundamental reason behind Saito's approach is the lack of perverse t-exactness of the internal tensor product: the bi-triangulated functor

$$- \otimes - : D_c^b(S, \mathbb{Q}) \times D_c^b(S, \mathbb{Q}) \to D_c^b(S, \mathbb{Q})$$

does not respect the perverse t-structure unless $\dim(S) = 0$. On the other hand, the external tensor product

$$- \boxtimes - : D_c^b(S_1, \mathbb{Q}) \times D_c^b(S_2, \mathbb{Q}) \to D_c^b(S_1 \times S_2, \mathbb{Q})$$

does always respect the perverse t-structures; it restricts to a bi-exact functor

$$- \boxtimes - : \mathrm{Perv}(S_1) \times \mathrm{Perv}(S_2) \to \mathrm{Perv}(S_1 \times S_2)$$

and, in fact, it can be canonically recovered from the latter modulo Beilinson's equivalences. In the setting of mixed Hodge modules, one is thus led to construct the external tensor product on the perverse hearts in the first place. (Note that what ultimately prevents the internal tensor product from being t-exact is the lack of t-exactness of the inverse image functor $\Delta_S^*$; however, it turns out that the latter can be conveniently expressed in terms of t-exact functors, which saves the day. We prefer to gloss over this point, as it will play no role in the present paper.) It is fair to say that Saito's impressive work deals mainly with the concrete definition of the single functors on mixed Hodge modules, leaving the most genuinely category-theoretic aspects of the construction aside. In particular, the associativity, symmetry, and unitarity properties of Saito's monoidal structure are not studied systematically in [Sai90]. Our work thus provides an adequate abstract framework for Saito's construction. Based on the same general framework, in [Ter24F] and [Ter24E] we further investigate more specific questions related to the monoidality of inverse image functors and the construction of unit constraints.

The same need for precise foundations for the theory of monoidal fibered categories based on the external tensor product is even more visible in the setting of *perverse Nori motives*, a further motivic refinement of Saito's coefficient system recently introduced by F. Ivorra and S. Morel in [IM24]. Again, the starting datum is a system of abelian categories of enhanced perverse sheaves, and the goal is to equip their bounded derived categories with the same functoriality as the underlying constructible derived categories. Contrary to the case of mixed Hodge modules, perverse Nori motives are defined by a purely categorical construction, which prevents one from describing their objects and morphisms in any concrete manner. In fact, the nature of Ivorra and Morel's categories makes it impossible to construct functors on them if not by lifting suitable t-exact functors on the constructible derived categories, as an abstract application of a certain categorical universal property. As long as one is concerned with the definition of the single functors, Saito's ideas can be successfully adapted to the motivic setting. However, making sure that these functors give rise to a coherent structure is a rather delicate question, for the structural reasons just mentioned. In our article [Ter24N], we apply the general method developed here to constructing the monoidal structure on perverse Nori motives, again starting from the external tensor product on the perverse hearts. In the motivic setting, the highly inexplicit nature of the categories involved forces us to be as formally precise as possible. For this reason, having a general dictionary expressing the usual structures and properties of the internal tensor product as structures and properties of the external tensor product is highly desirable in our opinion.

**Main results.** In order not to restrict the application range of our techniques unnecessarily, and also for sake of notational and conceptual clarity, we formulate our results in the general framework of fibered categories, thus avoiding any reference to perverse sheaves or to concrete geometric situations.
Given a small category $\mathcal{S}$ admitting binary products, we introduce the language of internal and external tensor structures on a given $\mathcal{S}$-fibered category $\mathcal{H}$. Our treatment of internal tensor structures is just a re- arrangement of the classical theory of monoidal $\mathcal{S}$-fibered categories: for example, we introduce associativity, commutativity, and unit constraints (as well as their compatibility conditions) as formalized in Mac Lane's classical paper [Mac63], and we describe their compatibility with inverse image functors as done in Ayoub's work [Ayo07a]. The only difference is that, in order to make the exposition as neat as possible, we prefer not to include an associativity constraint from the very beginning but rather treat it as an accessory structure, exactly as one sometimes does with commutativity and unit constraints. Our theory of external tensor structures should be new in the literature. The main notions that we introduce are the natural transliteration of the corresponding notions for internal tensor structures. This operation works nicely both for monoidality isomorphisms and for each of the three types of additional constraint (as well as for their compatibility conditions).

Motivated by the theory of perverse sheaves and by the abstract theory of six functor formalisms (as formalized in [Ayo07a, Ayo07b], [CD19] or [DG22]), we also study the so-called *projection formulae* in our setting of tensor structures. Not surprisingly, it turns out that the usual formulation in terms of the internal tensor product admits a natural translation in terms of the external tensor product. Again, the main difference with respect to the existing literature is that we treat the validity of projection formulae as a separate property rather than imposing it from the beginning.

Generally speaking, an important ingredient in our presentation is the notion of *equivalence* between internal or external tensor structures: this allows us to make it precise in which sense the classical formulae (2) and (3) should be considered to be mutually inverse. Our first main result can be stated as follows:

**Theorem (Theorem 6.1, Proposition 7.7).** Fix an $\mathcal{S}$-fibered category $\mathcal{H}$. Then there exist canonical mutually inverse bijections between equivalence classes of (unitary, symmetric, associative) internal and external tensor structures on $\mathcal{H}$. Moreover, these bijections respect the validity of projection formulae.

It is worth noting that all the equivalences between tensor structures involved in the proofs are explicit and canonical, so that the actual result is more precise than what asserted.

In the same spirit, given a morphism of $\mathcal{S}$-fibered categories $R : \mathbb{H}_1 \to \mathbb{H}_2$, we introduce the notion of internal and external tensor structures on $R$ (with respect to two given internal or external tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$, respectively). In the internal setting, we recover the usual notion of monoidal morphism between monoidal $\mathcal{S}$-fibered categories; in the external setting, we consider its natural transliteration. We then study the compatibility of internal and external tensor structures on $R$ with given associativity, commutativity, or unit constraints on $\mathbb{H}_1$ and $\mathbb{H}_2$. Our second main result can be stated as follows:

**Theorem (Theorem 8.1).** Fix a morphism of $\mathcal{S}$-fibered categories $R : \mathbb{H}_1 \to \mathbb{H}_2$. Then, under the bijective correspondence of the previous theorem, there exist canonical mutually inverse bijections between (unitary, symmetric, associative) internal and external tensor structures on $R$ with respect to two given (unitary, symmetric, associative) tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$.

In order for this statement to even make sense, one needs to use the explicit equivalences witnessing the bijections between internal and external tensor structures on $\mathbb{H}_1$ and on $\mathbb{H}_2$ as mentioned above.

Concretely, all the axioms introduced in the course of the paper ask for the commutativity of suitable natural diagrams; similarly, both the main results described above and all the intermediate results used to prove them amount to showing the commutativity of certain natural diagrams. Checking the commutativity of these diagrams directly is not hard: in fact, it does not require any new insight once the various claims are made explicit. The main issue is that one often has to decompose the diagrams under consideration into several pieces, and the resulting figures rarely fit into the space of one or two pages. As a partial workaround, we have chosen to include a few auxiliary results (Lemma 2.4, Lemma 2.8, Lemma 8.3 and
**Lemma 8.6)** which allow us to cleverly simplify the diagrams in question; in this way, it also becomes easier to understand the roles that the various axioms play in each proof.

These auxiliary results have independent interest, since they make it clear that internal and external tensor structures, both on fibered categories and on morphisms of such, are just suitable variations on the theme of fibered categories. For instance, we show that an internal tensor structure on a given $\mathcal{S}$-fibered category can be encoded into a plain fibered category over $\mathcal{S} \times \mathcal{I}$, where $\mathcal{I}$ denotes the category with two objects and one non-identity arrow (Construction 2.5); similarly, an external tensor structure can be encoded into a plain fibered category over $\mathcal{S}^2 \times \mathcal{I}$. One could certainly push this point of view further so as to include associativity, commutativity, and unit constraints; we leave this task to the interested reader.

**Related work.** The current literature counts several papers devoted to the foundations of monoidal fibered categories at various levels of abstraction. The interpretation of monoidal structures on $\mathcal{S}$-fibered categories in terms of categories fibered over a lager base discussed the present paper was influenced by F. Hörmann’s approach to six functor formalisms in the language of derivators, as developed in [Hör17]. The classical Grothendieck construction relating contravariant pseudo-functors with domain $\mathcal{S}$ to fibrations over $\mathcal{S}$ admits a monoidal analogue, investigated in [Shu08] and [MV20] in the 2-categorical setting and in [Ram22] in the $\infty$-categorical setting.

The present paper differs in spirit from these and similar works: our aim is not to encode the whole theory of monoidal fibered categories into a synthetic conceptual structure, but rather to turn the usual set of explicit axioms of the theory (given in terms of the internal tensor product) into a different but equivalent explicit set of axioms (tailored to the external tensor product). This means that our results can be used in practice to construct well-behaved monoidal structures on fibered categories in those contexts where it is more natural to work with the external than with the internal tensor product: one can construct associativity, commutativity, and unit constraints directly in terms of the external tensor product, and the coherence of the monoidal structure is guaranteed by an explicit list of natural conditions which often can be checked by hand. We believe that this is the main advantage offered by our approach.

**Structure of the paper.** Throughout the paper, we work over a fixed small category $\mathcal{S}$. Essentially, all fibered categories involved are fibered over $\mathcal{S}$; the only exception is represented by certain auxiliary fibered categories over larger bases (such as $\mathcal{S} \times \mathcal{I}$ or $\mathcal{S}^2 \times \mathcal{I}$, as already mentioned) which are not involved explicitly in the proofs of the main results.

In Section 1 we collect the general notions on fibered categories employed in the paper, including those of section, morphism, and image of a section under a morphism. We also prove a couple of useful auxiliary results (Lemma 1.3 and Lemma 1.8) which are repeatedly used in the sequel in order to shorten our constructions and simplify certain proofs.

Starting from Section 2, we assume that the base category $\mathcal{S}$ admits binary products: this is only needed to define the external tensor product; it is irrelevant for our discussion whether $\mathcal{S}$ possesses or not a terminal object. In Section 2 we explain how to rephrase the structure of a monoidal $\mathcal{S}$-fibered category purely in terms of the external tensor product. In order to highlight the symmetry between the internal and external versions of the tensor product, it is useful to write down the corresponding axioms of the two versions of the theory in parallel. To this end, we introduce appropriate notions of *internal tensor structure* and of *external tensor structure* (Definition 2.2(1) and Definition 2.6(1), respectively) that only include the single tensor product functors and their monoidality isomorphisms. We then explain how to transform internal tensor structures into external tensor structures (Lemma 2.10) and conversely (Lemma 2.11). Moreover, we introduce natural notions of *equivalence* between internal or external tensor structures (Definition 2.2(2) and Definition 2.6(2), respectively) that allow us to make it precise in what sense the two formulae (2) and (3) can be promoted to mutually inverse constructions (Proposition 2.12).
In Section 3, Section 4 and Section 5 we consider associativity, commutativity, and unit constraints, respectively. Following the approach of Section 2, we start by introducing the internal and external versions of the relevant constraint, and we refine the notions of equivalence of tensor structures accordingly; we then explain how to transform an internal constraint into an external one and conversely; lastly, we show that the canonical bijection between equivalence classes of internal and external tensor structures can be refined compatibly with the constraints under consideration.

In Section 6 we examine the mutual compatibility conditions between associativity, commutativity, and unit constraints. Again, each of these conditions can be formulated both in the internal and in the external setting, and the homologous conditions correspond to one another under the bijection between internal and external tensor structures. Once we have this last ingredient, we are able to deduce our first main result (Theorem 6.1) in its full strength.

The short Section 7 is devoted to the study of projection formulae, which are an important property of most monoidal fibered categories of geometric nature. We show that the usual projection formulae in the language of internal tensor structures have a natural analogue in the language of external tensor structures in such a way that the bijection of Proposition 2.12 respects the validity of projection formulae (Proposition 7.7).

In Section 8 we study monoidality of morphisms between monoidal $\mathcal{S}$-fibered categories. In the first half of the section, we introduce the notion of *internal tensor structure* and of *external tensor structure* on a morphism of $\mathcal{S}$-fibered categories with respect to given internal or external tensor structures on the two single $\mathcal{S}$-fibered categories involved (Definition 8.2 and Definition 8.5, respectively). By analogy with the case of monoidal structures on single $\mathcal{S}$-fibered categories, we explain how to transform internal tensor structures on a morphism into external tensor structures on the same morphism (Lemma 8.9) and conversely (Lemma 8.10); this is done coherently with the translation procedure between internal and external tensor structures on the two $\mathcal{S}$-fibered categories involved. Then we prove that these two constructions are mutually inverse modulo equivalence of internal and external tensor structures on the two sides (Proposition 8.11). In the second half of the section, we refine such a bijection with respect to associativity, commutativity, and unit constraints: to this end, we consider the natural properties of associativity, symmetry and unitarity for tensor structures on morphisms, both in the internal and in the external setting. We show without difficulty that the correspondence between internal and external tensor structures is compatible with each of these properties. Once we have this last ingredient, we are able to deduce our second main result (Theorem 8.1) in its full strength.

In the final Section 9, motivated by our applications to the setting of perverse sheaves, we specialize our discussion to the case of triangulated $\mathcal{S}$-fibered categories. We fully spell out the natural triangulated analogues of the various definitions and results obtained previously, but we omit all details of proof. The commutativity conventions that we have chosen here are coherent with the usual conventions for derived categories of abelian ($\mathcal{S}$-fibered) categories endowed with bi-exact tensor product functors.

**Acknowledgments.** The contents of this paper correspond to the first chapter of my Ph.D. thesis, written at the University of Freiburg under the supervision of Annette Huber-Klawitter. It is a pleasure to thank her for many useful discussions around the subject of this article, as well as for her constant support and encouragement. I would also like to thank Swann Tubach for some helpful notational comments on an earlier version of this work.

## NOTATION AND CONVENTIONS

* Unless otherwise dictated, categories are assumed to be small with respect to some fixed universe.

* Let $\mathcal{S}$ be a small category.

- The notation $S \in \mathcal{S}$ means that $S$ is an object of $\mathcal{S}$.
- A composable tuple in $\mathcal{S}$ is a tuple of the form $\underline{g} = (g_1, \dots, g_r)$ where $g_1, \dots, g_r$ are morphisms in $\mathcal{S}$ such that the composition $g_r \circ \dots \circ g_1$ makes sense in $\mathcal{S}$.

- We say that a sequence $\underline{g}^{(1)} = (g_1^{(1)}, \dots, g_{r_1}^{(1)})$, $\dots, \underline{g}^{(n)} = (g_1^{(n)}, \dots, g_{r_n}^{(n)})$ of composable tuples in $\mathcal{S}$ is composable if, for each $i = 1, \dots, n-1$, the composition $g_1^{(i+1)} \circ g_{r_i}^{(i)}$ makes sense in $\mathcal{S}$.

• Let $\mathcal{S}$ be a small category admitting binary products.

- Given two composable tuples in $\mathcal{S}$ with the same length

$$ \underline{g}^{(1)} = (g_1^{(1)}, \dots, g_r^{(1)}), \quad \underline{g}^{(2)} = (g_1^{(2)}, \dots, g_r^{(2)}), $$

we let $\underline{g}^{(1)} \times \underline{g}^{(2)}$ denote the composable tuple $(g_1^{(1)} \times g_1^{(2)}, \dots, g_r^{(1)} \times g_r^{(2)})$.

- For every object $S \in \mathcal{S}$ and every integer $n \ge 2$, we let

$$ \Delta_S^{(n)} : S \hookrightarrow S \times \dots \times S $$

denote the diagonal embedding of $S$ into the direct product of $n$ copies of $S$ in $\mathcal{S}$. In the special case where $n=2$, we omit the superscript from the notation.

- Given a collection of objects $S_{i_1}, \dots, S_{i_n} \in \mathcal{S}$ indexed by natural numbers $i_1, \dots, i_n$, for every substring $j_1 \dots j_r$ of the string $i_1 \dots i_n$ we let

$$ pr_{j_1 \dots j_r}^{i_1 \dots i_n} : S_{i_1} \times \dots \times S_{i_n} \to S_{j_1} \times \dots \times S_{j_r} $$

denote the canonical projection.

- Given a collection of objects $S_{i_1}, \dots, S_{i_n} \in \mathcal{S}$ indexed by a set of natural numbers $I = \{i_1, \dots, i_n\}$, for every permutation $\sigma \in \text{Sym}(I)$ we let

$$ \tau_{\sigma} : S_{i_1} \times \dots S_{i_n} \xrightarrow{\sim} S_{\sigma(i_1)} \times \dots \times S_{\sigma(i_n)} $$

denote the canonical permutation isomorphism. In the special case where $n=2$ and $\sigma$ is the non-identity permutation, we omit the subscript from the notation.

• Given categories $\mathcal{C}_1, \dots, \mathcal{C}_n$, we let $\mathcal{C}_1 \times \dots \times \mathcal{C}_n$ denote their direct product category.

# 1. RECOLLECTIONS ON FIBERED CATEGORIES

Throughout this paper, we fix a small category $\mathcal{S}$; starting from Section 2 we will need to assume binary products to exist in $\mathcal{S}$, but this is not necessary for the moment. All fibered categories mentioned in this paper are fibered over $\mathcal{S}$ unless stated otherwise.

The conventions on 2-categories and 2-functors that we follow are those adopted in [Ayo07a, § 1.4.1] and in [CD19, § 1.1]: what we call a fibered category here is what in the original reference [SGA1, Exp. VI, Defn. 7.1] is called a "catégorie clivée normalisée"; we consider 2-categories in the strict sense but 2-functors in the lax sense. See [Mac71, § XII.3] for generalities on 2-categories.

## 1.1. Fibered categories and sections.
In the first part of this section, we quickly review $\mathcal{S}$-fibered categories. We define an $\mathcal{S}$-fibered category as a contravariant 2-functor from $\mathcal{S}$ to the 2-category of small categories. For simplicity, we assume every such 2-functor to be strictly unital; in fact, this is not a severe requirement (see for example [Del01, Lemma 2.5]). In detail:

**Definition 1.1.** An $\mathcal{S}$-fibered category $\mathcal{H}$ is the datum of

• for every $S \in \mathcal{S}$, a category $\mathcal{H}(S)$,

• for every morphism $f : T \to S$ in $\mathcal{S}$, a functor

$$ f^* : \mathcal{H}(S) \to \mathcal{H}(T), $$

called the inverse image functor along $f$,
• for every pair of composable morphisms $f: T \to S$ and $g: S \to V$ in $\mathcal{S}$, a natural isomorphism of functors $\mathbb{H}(V) \to \mathbb{H}(T)$

$$ (4) \qquad \mathrm{conn} = \mathrm{conn}_{f,g} : (gf)^*A \xrightarrow{\sim} f^*g^*A $$

called the *connection isomorphism* at $(f,g)$

such that the following conditions are satisfied:

(\mathcal{S}-fib-0) For every $S \in \mathcal{S}$, we have $\mathrm{id}_S^* = \mathrm{id}_{\mathbb{H}(S)}$.

(\mathcal{S}-fib-1) For every triple of composable morphisms $f: T \to S$, $g: S \to V$ and $h: V \to W$ in $\mathcal{S}$, the diagram of functors $\mathbb{H}(W) \to \mathbb{H}(T)$

$$ (5) \qquad \begin{array}{ccc} (hgf)^*A & \xrightarrow{\mathrm{conn}_{f,hg}} & f^*(hg)^*A \\ \Big\downarrow^{\mathrm{conn}_{g,f,h}} & & \Big\downarrow^{\mathrm{conn}_{g,h}} \\ (gf)^*h^*A & \xrightarrow{\mathrm{conn}_{f,g}} & f^*g^*h^*A \end{array} $$

is commutative.

**Notation 1.2.** Given a composable tuple $g = (g_1, \dots, g_r)$ in $\mathcal{S}$, we let $g^*$ denote the composite of inverse image functors $g_1^* \circ \cdots \circ g_r^*$ in any $\mathcal{S}$-fibered category. In order to avoid possible confusion, we agree that giving an empty composable tuple $g$ just means giving an object $S \in \mathcal{S}$; in this case, we set $g^* := \mathrm{id}_S^* = \mathrm{id}_{\mathbb{H}(S)}$.

The following result explains the full meaning of axiom (\mathcal{S}-fib-1); it will serve as a model for similar results obtained in Section 2 and Section 8.

**Lemma 1.3.** Let $\mathbb{H}$ be a $\mathcal{S}$-fibered category, and fix a morphism $f: T \to S$ in $\mathcal{S}$. For every composable tuple in $\mathcal{S}$

$$ \underline{g} = (g_1, \dots, g_r) $$

satisfying $g_r \circ \dots \circ g_1 = f$, consider the functor $F_{\underline{g}}: \mathbb{H}(S) \to \mathbb{H}(T)$ defined by the formula

$$ F_{\underline{g}}(A) := \underline{g}^*A. $$

Then, given any two tuples $\underline{g}^{(1)}, \underline{g}^{(2)}$ as above, all natural isomorphisms of functors $\mathbb{H}(S) \to \mathbb{H}(T)$

$$ F_{\underline{g}^{(1)}}(A) \xrightarrow{\sim} F_{\underline{g}^{(2)}}(A) $$

obtained by composing connection isomorphisms (and inverses thereof) coincide.

*Proof.* It suffices to show that, for every tuple $\underline{g}$ as in the statement, every natural isomorphism of functors $\mathbb{H}(S) \to \mathbb{H}(T)$

$$ \phi : F_{\underline{g}}(A) \xrightarrow{\sim} f^*A $$

obtained by composing connection isomorphisms (and inverses thereof) coincides with the composite of inverse connection isomorphisms

$$ \alpha_{\underline{g}} : F_{\underline{g}}(A) := g_1^* \cdots g_r^* A \xleftarrow{\mathrm{conn}} (g_2 g_1)^* \cdots g_r^* \xleftarrow{\mathrm{conn}} \cdots \xleftarrow{\mathrm{conn}} (g_r \cdots g_1)^* A = f^*A. $$

To see this, write $\phi$ explicitly as

$$ \phi : F_{\underline{g}^{(0)}} \xrightarrow{\phi_1} F_{\underline{g}^{(1)}} \xrightarrow{\phi_2} \dots \xrightarrow{\phi_N} F_{\underline{g}^{(N)}} $$

where we set $\underline{g}^{(0)} := \underline{g}$, $\underline{g}^{(N)} := (f)$, and each factor $\phi_i$ is either a direct or an inverse connection isomorphism; note that, by construction, the number of inverse minus the number of direct connection isomorphisms among the $\phi_i$'s equals $r-1$. We prove our claim by analyzing the following cases separately:
* Suppose that all factors $\phi_i$ are inverse connection isomorphisms. Then, by repeated applications of axiom (S-fib-1), the claim follows by induction on $N = r - 1$.

* Suppose that there exists an index $n$ with $0 < n < N$ such that $\phi_i$ is a direct connection isomorphism for $1 \le i \le n$ and an inverse connection isomorphism for $n + 1 \le i \le N$. Then we can consider the factorization

$$ \phi : F_{\underline{g}^{(0)}} \xrightarrow{\psi_1} F_{\underline{g}^{(n)}} \xrightarrow{\psi_2} F_{\underline{g}^{(N)}} $$

where $\psi_1 := \phi_n \circ \cdots \circ \phi_1$ and $\psi_2 := \phi_N \circ \cdots \circ \phi_{n+1}$. By the previous case, we know that every natural isomorphism

$$ F_{\underline{g}^{(n)}}(A) \xrightarrow{\sim} F_{\underline{g}^{(N)}}(A) := f^*A $$

obtained as the composite of inverse connection isomorphisms coincides with $\alpha_{\underline{g}^{(n)}}$. In particular, we have

$$ \psi_2 = \alpha_{\underline{g}^{(n)}} = \alpha_{\underline{g}^{(0)}} \circ \psi_1^{-1}. $$

The claim follows from this, since we have the equality

$$ \phi = \psi_2 \circ \psi_1 = \alpha_{\underline{g}^{(n)}} \circ \psi_1 = \alpha_{\underline{g}^{(0)}}. $$

* Suppose only that at least one of the factors $\phi_i$ is a direct connection isomorphism. We want to transform this factorization into one of the form considered in the previous case. To this end it suffices to show that, given two consecutive factors $\phi_i$ and $\phi_{i+1}$ with $\phi_i$ inverse and $\phi_{i+1}$ direct, it is possible to write the composite $\phi_{i+1} \circ \phi_i$ in the form $\phi'_{i+1} \circ \phi'_i$ with $\phi'_i$ direct and $\phi'_{i+i}$ inverse. We prove this by distinguishing the following cases:

- If $\phi_i$ and $\phi_{i+1}$ are mutually inverse, they can be just cancelled out.

- If $\phi_i$ and $\phi_{i+1}$ are not mutually inverse and the sets of morphisms where they act are disjoint from one another, then the corresponding natural transformations can be interchanged.

- If $\phi_i$ and $\phi_{i+1}$ are not mutually inverse and the sets of morphisms where they act are not disjoint, then they fit into the bottom-right corner of a square of type (5). By axiom (S-fib-1) we can replace them with the edges in the top-left corner of the same square.

Therefore the claim follows from the previous case.

This concludes the proof.

□

Let us briefly review sections of S-fibered categories: this notion will play a role in the discussion about unit constraints in Section 5 and in the final part of Section 8.

**Definition 1.4.** Let $\mathbb{H}$ be an S-fibered category.

(1) A section $A$ of $\mathbb{H}$ over $\mathcal{S}$ is the datum of

* for every $S \in \mathcal{S}$, an object $A_S \in \mathbb{H}(S)$,

* for every morphism $f : T \to S$ in $\mathcal{S}$, an isomorphism in $\mathbb{H}(T)$

$$ A^* = A_f^* : f^*A_S \xrightarrow{\sim} A_T $$

satisfying the following condition:

(S-sect) For every pair of composable morphisms $f : T \to S$, $g : S \to V$ in $\mathcal{S}$, the diagram in $\mathbb{H}(T)$

$$ f^* g^* A_V \xrightarrow{A_g^*} f^* A_S $$

$$ \xrightarrow{\text{conn}_{f,g}} \xrightarrow{A_f^*} $$

$$ (gf)^* A_V \xrightarrow{A_{gf}^*} A_T $$
is commutative.

(2) Let $A$ and $B$ be two sections of $\mathbb{H}$. A morphism of sections $w : A \to B$ is the datum of
• for every $S \in \mathcal{S}$, a morphism

$w_S : A_S \to B_S$

satisfying the following condition:

(mor-$\mathcal{S}$-sect) For every morphism $f : T \to S$ in $\mathcal{S}$, the diagram in $\mathbb{H}(T)$

$$
\begin{tikzcd}
& & f^*A_S \arrow[r, "w_S"] \arrow[d, "A_f^*"'] & f^*B_S \arrow[d, "B_f^*"'] \\
A_T \arrow[r, "w_T"'] & B_T & &
\end{tikzcd}
$$

is commutative.

Remark 1.5. There are natural notions of identity and of composition for morphisms between sections of a same $\mathcal{S}$-fibered categories $\mathbb{H}$. Of course, a morphism of sections $w : A \to B$ is invertible with respect to this composition if and only if each morphism $w_S : A_S \to B_S$ is an isomorphism in $\mathbb{H}(S)$ for every $S \in \mathcal{S}$: this yields the notion of *isomorphism of sections*.

1.2. **Morphisms of fibered categories.** In the second part of this section, we quickly review morphisms of $\mathcal{S}$-fibered categories. Let $\mathbb{H}_1$ and $\mathbb{H}_2$ be two $\mathcal{S}$-fibered categories as in **Definition 1.1**; for every pair of composable pair $(f,g)$ in $\mathcal{S}$, we write the connection isomorphism at $(f,g)$ for $\mathbb{H}_1$ and $\mathbb{H}_2$ as $\mathrm{conn}_{f,g}^{(1)}$ and $\mathrm{conn}_{f,g}^{(2)}$, respectively.

**Definition 1.6.** (1) A morphism of $\mathcal{S}$-fibered categories $R : \mathbb{H}_1 \to \mathbb{H}_2$ is the datum of
• for every $S \in \mathcal{S}$, a functor

$R_S : \mathbb{H}_1(S) \to \mathbb{H}_2(S),$

• for every morphism $f : T \to S$ in $\mathcal{S}$, a natural isomorphism of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(T)$

$$
\theta = \theta_f : f^* R_S(A) \xrightarrow{\sim} R_T(f^* A),
$$

called the *R-transition isomorphism* along *f*

such that the following condition is satisfied:

(mor-$\mathcal{S}$-fib) For every pair of composable morphisms $f : T \to S$ and $g : S \to V$ in $\mathcal{S}$, the diagram of functors $\mathbb{H}_1(V) \to \mathbb{H}_2(T)$

$$
\begin{tikzcd}
& & (gf)^*R_V(A) \arrow[r, "\theta_{gf}"] & R_T((gf)^*A) \arrow[d, "\mathrm{conn}_{f,g}^{(2)}"'] \\
& \mathrm{conn}_{f,g}^{(2)} & & \\
f^*g^*R_V(A) \arrow[r, "\theta_g"] & f^*R_S(g^*A) \arrow[r, "\theta_f"] & R_T(f^*g^*A) &
\end{tikzcd}
$$

is commutative.

(2) If we are already given a family of functors $\mathcal{R} := \{R_S : \mathbb{H}_1(S) \to \mathbb{H}_2(S)\}_{S \in \mathcal{S}}$, we say that a collection of natural isomorphisms $\theta = \{\theta_f : f^* \circ R_S \xrightarrow{\sim} R_T \circ f^*\}_{f:T \to S}$ satisfying condition (mor-$\mathcal{S}$-fib) defines an $\mathcal{S}$-structure on the family $\mathcal{R}$.

Remark 1.7. Fix $S \in \mathcal{S}$. In view of axiom (\mathcal{S}-fib-0) from Definition 1.1, condition (mor-\mathcal{S}-fib) in the case where $f = g = \mathrm{id}_S$ means that the natural isomorphism of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(S)$

$$
\theta_{\mathrm{id}_S} : R_S(A) \xrightarrow{\sim} R_S(A)
$$

is idempotent. Hence it must be the identity.
The full meaning of axiom (mor-$\mathcal{S}$-fib) is explained by the following result, which is the generalization of Lemma 1.3 to morphisms of $\mathcal{S}$-fibered categories:

**Lemma 1.8.** Let $R: \mathbb{H}_1 \to \mathbb{H}_2$ be a morphism of $\mathcal{S}$-fibered categories, and fix a morphism $f: T \to S$ in $\mathcal{S}$. For every composable pair of composable tuples in $\mathcal{S}$

$$ \underline{g} = (g_1, \dots, g_r), \quad \underline{h} = (h_1, \dots, h_s) $$

satisfying $h_s \circ \dots \circ h_1 \circ g_r \circ \dots \circ g_1 = f$, consider the functor $\mathbb{H}_1(S) \to \mathbb{H}_2(T)$

$$ F_{(\underline{g}, \underline{h})}^{R}(A) := \underline{g}^{*} R_{T'}(\underline{h}^{*} A) $$

where $T'$ denotes the domain of $h_1$. Then, given any two pairs of tuples $(\underline{g}^{(1)}, \underline{h}^{(1)})$ and $(\underline{g}^{(2)}, \underline{h}^{(2)})$ as above, all natural isomorphisms of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(T)$

$$ F_{(\underline{g}^{(1)}, \underline{h}^{(1)})}^{R}(A) \xrightarrow{\sim} F_{(\underline{g}^{(2)}, \underline{h}^{(2)})}^{R}(A) $$

obtained by composing connection isomorphisms and $R$-transition isomorphisms (and inverses thereof) co-incide.

It would be possible to prove this result via a detailed case-by-case analysis as done for Lemma 1.3; however, the mixture of connection isomorphisms and $R$-transition isomorphisms would force us to consider more cases and introduce more reduction steps in the argument. We prefer to introduce an auxiliary construction which will reduce us directly to the situation of Lemma 1.3, thereby avoiding all additional difficulties. To this end, we need to introduce some more notation:

**Notation 1.9.** • We let $\mathcal{I}$ be the category with two objects, denoted 1 and 2, and exactly one non-identity arrow $r: 2 \to 1$.

• Given a small category $\mathcal{S}$, we consider the product category $\mathcal{S} \times \mathcal{I}$. We write an object $S \in \mathcal{S} \times \mathcal{I}$ as a pair $(S; i)$ with $S \in \mathcal{S}$ and $i \in \{1, 2\}$. For every $S \in \mathcal{S}$, we let $r_S$ denote the morphism $(\mathrm{id}_S, r) : (S; 2) \to (S; 1)$. Note that there exists a natural partition of morphisms in $\mathcal{S} \times \mathcal{I}$ into the following three families, each of which is parameterized by the family of all morphisms $f: T \to S$ in $\mathcal{S}$:

– morphisms of the form $(f; 1) : (T; 1) \to (S; 1)$;

– morphisms of the form $(f; 2) : (T; 2) \to (S; 2)$;

– morphisms of the form $r_S \circ (f; 2) : (T; 2) \to (S; 1)$.

We will always use this presentation for morphisms in $\mathcal{S} \times \mathcal{I}$.

**Construction 1.10.** Let $R: \mathbb{H}_1 \to \mathbb{H}_2$ be a morphism of $\mathcal{S}$-fibered categories. Then, following Notation 1.9, we can define an $(\mathcal{S} \times \mathcal{I})$-fibered category $\mathbb{H}_1 \coprod_R \mathbb{H}_2$ out of it, as follows:

• For every $(S; i) \in \mathcal{S} \times \mathcal{I}$, we set $(\mathbb{H}_1 \coprod_R \mathbb{H}_2)(S; i) := \mathbb{H}_i(S)$.

• Given a morphism $\phi$ in $\mathcal{S} \times \mathcal{I}$, we define the functor $\phi^*$ according to the type of $\phi$:

– if $\phi = (f; 1) : (T; 1) \to (S; 1)$, we set $\phi^*A := f^*A$;

– if $\phi = (f; 2) : (T; 2) \to (S; 2)$, we set $\phi^*B := f^*B$;

– if $\phi = r_S \circ (f; 2)$, we set $\phi^*A := f^*R_S(A)$.

• Given two composable morphisms $\phi$ and $\psi$ in $\mathcal{S} \times \mathcal{I}$, we define the connection isomorphism $\mathrm{conn}_{\phi, \psi}$ according to the type of $\phi$ and $\psi$:

(i) if $\phi = (f; 1)$ and $\psi = (g; 1)$, we set

$$ \mathrm{conn}_{\phi, \psi}(A) : (gf)^*A \xrightarrow{\mathrm{conn}_{f,g}^{(1)}} f^*g^*A; $$
(ii) if $\phi = (f; 2)$ and $\psi = (g; 2)$, we set

$$ \mathrm{conn}_{\phi, \psi}(B) : (gf)^*B \xrightarrow{\mathrm{conn}_{f,g}^{(2)}} f^*g^*B; $$

(iii) if $\phi = (f; 2)$ and $\psi = r_V \circ (g; 2)$, we set

$$ \mathrm{conn}_{\phi, \psi}(A) : (gf)^*R_V(A) \xrightarrow{\mathrm{conn}_{f,g}^{(2)}} f^*g^*R_V(A); $$

(iv) if $\phi = r_S \circ (f; 2)$ and $\psi = (g; 1)$, we set

$$ \mathrm{conn}_{\phi, \psi} : (gf)^*R_S(A) \xrightarrow{\mathrm{conn}_{f,g}^{(2)}} f^*g^*R_V(A) \xrightarrow{\theta_g} f^*R_S(g^*A). $$

Let us check that this really defines an $(\mathcal{S} \times \mathcal{I})$-fibered category. It is clear that condition $((\mathcal{S} \times \mathcal{I})$-fib-0) is satisfied. We need to check that condition $((\mathcal{S} \times \mathcal{I})$-fib-1) holds as well: given three composable morphisms $\phi, \psi, \xi$ in $\mathcal{S} \times \mathcal{I}$, we have to show that the diagram

$$ \begin{array}{ccc} (\xi\psi\phi)^*A & \xrightarrow{\mathrm{conn}_{\phi, \xi\psi}} & \phi^*(\xi\psi)^*A \\ \Big\downarrow\mathrm{conn}_{\phi, \xi\psi} & & \Big\downarrow\mathrm{conn}_{\psi, \xi} \\ (\psi\phi)^*\xi^*A & \xrightarrow{\mathrm{conn}_{\phi, \psi}} & \phi^*\psi^*\xi^*A \end{array} $$

is commutative. We have the following possibilities, parametrized by triples of composable morphisms $f: T \to S$, $g: S \to V$ and $h: V \to W$ in $\mathcal{S}$:

(1) If $\phi = (f; 1)$, $\psi = (g; 1)$ and $\xi = (h; 1)$, we obtain the diagram

$$ \begin{array}{ccc} (hgf)^*A & \xrightarrow{\mathrm{conn}_{f,hg}^{(1)}} & f^*(hg)^*A \\ \Big\downarrow\mathrm{conn}_{g,f,h}^{(1)} & & \Big\downarrow\mathrm{conn}_{g,h}^{(1)} \\ (gf)^*h^*A & \xrightarrow{\mathrm{conn}_{f,g}^{(1)}} & f^*g^*h^*A \end{array} $$

which is commutative by axiom $(\mathcal{S}$-fib-1).

(2) If $\phi = (f; 2)$, $\psi = (g; 2)$ and $\xi = (h; 2)$, we obtain the diagram

$$ \begin{array}{ccc} (hgf)^*B & \xrightarrow{\mathrm{conn}_{f,hg}^{(2)}} & f^*(hg)^*B \\ \Big\downarrow\mathrm{conn}_{g,f,h}^{(2)} & & \Big\downarrow\mathrm{conn}_{g,h}^{(2)} \\ (gf)^*h^*B & \xrightarrow{\mathrm{conn}_{f,g}^{(2)}} & f^*g^*h^*B \end{array} $$

which is commutative by axiom $(\mathcal{S}$-fib-1).

(3) If $\phi = (f; 2)$, $\psi = (g; 2)$ and $\xi = r_W \circ (h; 2)$, we obtain the diagram

$$ \begin{array}{ccc} (hgf)^*R_W(A) & \xrightarrow{\mathrm{conn}_{f,hg}^{(2)}} & f^*(hg)^*R_W(A) \\ \Big\downarrow\mathrm{conn}_{g,f,h}^{(2)} & & \Big\downarrow\mathrm{conn}_{g,h}^{(2)} \\ (gf)^*h^*R_W(A) & \xrightarrow{\mathrm{conn}_{f,g}^{(2)}} & f^*g^*h^*R_W(A) \end{array} $$

which is again commutative by axiom $(\mathcal{S}$-fib-1).
(4) If $\phi = (f; 2)$, $\psi = r_V \circ (g; 2)$ and $\xi = (h; 1)$, we obtain the outer part of the diagram

<figure><img src="image_1.png" /></figure>

where the upper square is commutative by axiom (S-fib-1) and the lower square is commutative by naturality.

(5) Finally, if $\phi = r_S \circ (f; 2)$, $\psi = (g; 1)$ and $\xi = (h; 1)$, we obtain the outer part of the diagram

<figure><img src="image_2.png" /></figure>

where the upper-left piece is commutative by axiom (S-fib-1), the lower-left piece is commutative by naturality, and the right-most piece is commutative by axiom (mor-S-fib).

In fact, the argument shows that giving a morphism of S-fibered categories $R: \mathbb{H}_1 \to \mathbb{H}_2$ is the same as giving an (S × I)-fibered category $\mathbb{H}$ satisfying $\mathbb{H}|_{(S; i)} = \mathbb{H}_i$ for $i = 1, 2$ and $r_S^* = R_S$ for all $S \in S$.

*Proof of Lemma 1.8.* As a consequence of Construction 1.10, this is a particular case of Lemma 1.3 over the base category $S \times I$. $\square$

We conclude this preliminary discussion by reviewing the notion of image of a section along a morphism of S-fibered categories.

**Construction 1.11.** Let $R: \mathbb{H}_1 \to \mathbb{H}_2$ be a morphism of S-fibered categories, and let $A$ be a section of $\mathbb{H}_1$ over $S$. We define the image of $A$ under $R$ to be the section $R(A)$ of $\mathbb{H}_2$ over $S$ defined as follows:

* for every $S \in S$, the object $R(A)_S \in \mathbb{H}_2(S)$ is just $R_S(A_S)$;

* for every morphism $f: T \to S$ in $S$, the isomorphism in $\mathbb{H}_2(T)$

$$R(A)_f^* : f^*R(A)_S \xrightarrow{\theta_f} R(A)_T$$

is the composite

$$f^*R_S(A_S) \xrightarrow{\theta_f} R_T(f^*A_S) \xrightarrow{\sim} R_T(A_T).$$

Let us check that this definition indeed satisfies condition (S-sect): given two composable morphisms $f: T \to S$ and $g: S \to V$ in $S$, we have to show that the diagram in $\mathbb{H}_2(T)$

$$\begin{tikzcd}
(gf)^*R(A)_V \arrow[r, "R(A)_{gf}^*"'] & R(A)_T \arrow[d, "\text{conn}_{f,g}^{(2)}"'] \\
f^*g^*R(A)_V \arrow[r, "R(A)_g^*"'] & f^*R(A)_S \arrow[r, "R(A)_f^*"'] & R(A)_T
\end{tikzcd}$$
is commutative. Expanding the definitions, we obtain the outer part of the diagram

<figure><img src="image_3.png" /></figure>

where the lower central piece is commutative by naturality, the left-most piece is commutative by axiom (mor- *S*-fib), and the right-most piece is commutative by axiom (*S*-sect). This proves the claim.

## 2. INTERNAL AND EXTERNAL TENSOR STRUCTURES

From now henceforth, we assume that the small category *S* admits binary products; in most concrete applications *S* also happens to possess a terminal object, but this is not necessary for our constructions.

From now until the end of Section 7 we also fix an *S*-fibered category $\mathbb{H}$. In the present section, we start our investigations about monoidal *S*-fibered categories by introducing the notions of internal and external tensor structure on $\mathbb{H}$ together with the corresponding notions of equivalence.

**Notation 2.1.** From now on, we simplify the notation by systematically adopting the following conventions:

* We write all connection isomorphisms of *S*-fibered categories as equalities (this is legitimate in view of Lemma 1.3).

* We write all transition isomorphisms $\theta_f$ associated to a morphism of *S*-fibered categories simply as $\theta$. More generally, when a structure on *S*-fibered category is defined by a family of natural isomorphisms parameterized by objects or morphisms of *S*, we often indicate all members of this family by the same symbol, without mentioning the specific parameter involved. In particular, this will be applied to all the notation concerning tensor structures.

**2.1. Tensor structures and their equivalences.** To begin with, we introduce internal and external tensor structures and equivalences thereof. In order to keep the exposition as neat as possible, it is convenient to start from the coarsest possible kind of structure including only monoidality isomorphisms: in particular, we do not include associativity constraints in the picture from the beginning but rather regard them as one possible additional structure (on the same level as commutativity and unit constraints) and treat them separately in Section 3.

**Definition 2.2.** (1) An internal tensor structure $(\otimes, m)$ on $\mathbb{H}$ is the datum of

* for every $S \in \mathcal{S}$, a functor

$$ - \otimes - = - \otimes_S - : \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S), $$

called the *internal tensor product functor* over $S$,

* for every morphism $f: T \to S$ in $\mathcal{S}$, a natural isomorphism of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ m = m_f : f^* A \otimes f^* B \xrightarrow{\sim} f^*(A \otimes B), $$

called the *internal monoidality isomorphism* along $f$ satisfying the following condition:
(mITS) For every pair of composable morphisms $f: T \to S$ and $g: S \to V$ in $\mathcal{S}$, the diagram of functors $\mathbb{H}(V) \times \mathbb{H}(V) \to \mathbb{H}(T)$

$$ \begin{array}{ccc} (gf)^* A \otimes (gf)^* B & \xrightarrow{m} & (gf)^*(A \otimes B) \\ \Vert & & \Vert \\ f^*g^* A \otimes f^*g^* B & \xrightarrow{m} & f^*(g^*A \otimes g^*B) \xrightarrow{m} f^*g^*(A \otimes B) \end{array} $$

is commutative.

(2) Let $(\otimes, m)$ and $(\otimes', m')$ be two internal tensor structures on $\mathbb{H}$. An equivalence of internal tensor structures $e: (\otimes, m) \to (\otimes', m')$ is the datum of

* for every $S \in \mathcal{S}$, a natural isomorphism of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ e = e_S : A \otimes B \xrightarrow{\sim} A \otimes' B $$

satisfying the following condition:

(eq-ITS) For every morphism $f: T \to S$ in $\mathcal{S}$, the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ \begin{array}{ccc} f^* A \otimes f^* B & \xrightarrow{m} & f^*(A \otimes B) \\ \searrow & & \searrow \\ f^* A \otimes' f^* B & \xrightarrow{m'} & f^*(A \otimes' B) \end{array} $$

is commutative.

Remark 2.3. Fix $S \in \mathcal{S}$. In view of axiom (S-fib-0) of Definition 1.1, condition (mITS) in the case where $f = g = \mathrm{id}_S$ means that the natural isomorphism of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ m_{\mathrm{id}_S} : A \otimes B \xrightarrow{\sim} A \otimes B $$

is idempotent. Hence it must be the identity.

The following result clarifies the meaning of axiom (mITS): it guarantees that, between any two given iterations of tensor product functors and inverse image functors, it is possible to construct at most one natural isomorphism by combining connection isomorphisms and internal monoidality isomorphisms in any possible order. As the reader might expect, this is just another variant of Lemma 1.3.

**Lemma 2.4.** Let $(\otimes, m)$ be an internal tensor structure on $\mathbb{H}$, and let $f: T \to S$ be a morphism in $\mathcal{S}$. For every composable pair of composable tuples in $\mathcal{S}$

$$ \underline{g} = (g_1, \dots, g_r), \quad \underline{h} = (h_1, \dots, h_s) $$

satisfying $h_s \circ \dots \circ h_1 \circ g_r \circ \dots \circ g_1 = f$, consider the functor $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ F_{(g,h)}^{\otimes}(A, B) := \underline{g}^*(\underline{h}^* A \otimes \underline{h}^* B). $$

Then, given any two pairs of tuples $(\underline{g}^{(1)}, \underline{h}^{(1)})$ and $(\underline{g}^{(2)}, \underline{h}^{(2)})$ as above, all possible isomorphisms of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ F_{(g^{(1)}, \underline{h}^{(1)})(A, B)}^{\otimes} \xrightarrow{\sim} F_{(g^{(2)}, \underline{h}^{(2)})(A, B)}^{\otimes} $$

obtained by composing connection isomorphisms and internal monoidality isomorphisms (and inverses thereof) coincide.

It would be possible to prove this result directly; however, the argument would be quite cumbersome due to the mixture of connection isomorphisms and internal monoidality isomorphisms. We prefer to reduce ourselves directly to the setting of usual fibered categories considered in Section 1 via the following auxiliary construction:
(3) If $\phi = (f; 2)$, $\phi = (g; 2)$ and $\xi = r_W \circ (h; 2)$, we obtain the diagram

$$ \begin{array}{ccc} (hgf)^*(A \otimes B) & = & f^*(hg)^*(A \otimes B) \\ \parallel & & \parallel \\ (gf)^*h^*(A \otimes B) & = & f^*g^*h^*(A \otimes B) \end{array} $$

which is again commutative by axiom (S-fib-1).

(4) If $\phi = (f; 2)$, $\psi = r_V \circ (g; 2)$ and $\xi = (h; 1)$, we obtain (the outer rectangle of) the diagram

$$ \begin{array}{ccc} (hgf)^*(A \otimes B) & = & f^*(hg)^*(A \otimes B) \\ \parallel & & \parallel \\ (gf)^*h^*(A \otimes B) & = & f^*g^*h^*(A \otimes B) \\ \uparrow & & \uparrow \\ (gf)^*(h^*A \otimes h^*B) & = & f^*g^*(h^*A \otimes h^*B) \end{array} $$

where the upper square is commutative by axiom (S-fib-1) and the lower square is commutative by naturality.

(5) If $\phi = r_S \circ (f; 2)$, $\psi = (g; 1)$ and $\xi = (h; 1)$, we obtain (the outer rectangle of) the diagram

$$ \begin{array}{ccc} (hgf)^*(A \otimes B) & = & f^*(hg)^*(A \otimes B) \leftarrow_m f^*((hg)^*A \otimes (hg)^*B) \\ \parallel & & \parallel \\ (gf)^*h^*(A \otimes B) & = & f^*g^*h^*(A \otimes B) \\ \uparrow & & \uparrow \\ (gf)^*(h^*A \otimes h^*B) & = & f^*g^*(h^*A \otimes h^*B) \leftarrow_m f^*(g^*h^*A \otimes g^*h^*B) \end{array} $$

where the upper-left piece is commutative by axiom (S-fib-1), the lower-left piece is commutative by naturality, and the right-most piece is commutative by axiom (mITS).

*Proof of Lemma 2.4.* As a consequence of Construction 2.5, this is just a particular case of Lemma 1.3. $\square$

In a similar way, we now introduce external tensor structures and equivalences thereof.

**Definition 2.6.** (1) An *external tensor structure* $(\boxtimes, m)$ on $\mathbb{H}$ is the datum of

* for every $S_1, S_2 \in \mathcal{S}$, a functor

$$ - \boxtimes - = - \boxtimes_{S_1, S_2} : \mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2), $$

called the *external tensor product functor* over $(S_1, S_2)$,

* for every choice of morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, $i = 1, 2$, a natural isomorphism of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(T_1 \times T_2)$

$$ m = m_{f_1, f_2} : f_1^* A_1 \boxtimes f_2^* A_2 \xrightarrow{\sim} (f_1 \times f_2)^*(A_1 \boxtimes A_2), $$

called the *external monoidality isomorphism* along $(f_1, f_2)$

satisfying the following condition:
(mETS) For every choice of composable morphisms $f_i : T_i \to S_i$, $g_i : S_i \to V_i$, $i = 1, 2$, the diagram of functors $\mathbb{H}(V_1) \times \mathbb{H}(V_2) \to \mathbb{H}(T_1 \times T_2)$

$$ (g_1 f_1)^* A_1 \boxtimes (g_2 f_2)^* A_2 \xrightarrow[m]{\phantom{\simeq}} (g_1 f_1 \times g_2 f_2)^* (A_1 \boxtimes A_2) \\ \| \\ f_1^* g_1^* A_1 \boxtimes f_2^* g_2^* A_2 \xrightarrow[m]{\phantom{\simeq}} (f_1 \times f_2)^* (g_1^* A_1 \boxtimes g_2^* A_2) \xrightarrow[m]{\phantom{\simeq}} (f_1 \times f_2)^* (g_1 \times g_2)^* (A_1 \boxtimes A_2) $$

is commutative.

(2) Let $(\boxtimes', m')$ and $(\boxtimes, m)$ be two external tensor structures on $\mathbb{H}$. An equivalence of external tensor structures $e : (\boxtimes', m') \to (\boxtimes, m)$ is the datum of

• for every $S_1, S_2 \in \mathcal{S}$, a natural isomorphism of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ e = e_{S_1, S_2} : A_1 \boxtimes' A_2 \xrightarrow{\sim} A_1 \boxtimes A_2 $$

satisfying the following condition:

(eq-ETS) For every pair of morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(T_1 \times T_2)$

$$ \begin{array}{ccc} f_1^* A_1 \boxtimes' f_2^* A_2 & \xrightarrow{m'} & (f_1 \times f_2)^* (A_1 \boxtimes' A_2) \\ \downarrow^e & & \downarrow^e \\ f_1^* A_1 \boxtimes f_2^* A_2 & \xrightarrow{m} & (f_1 \times f_2)^* (A_1 \boxtimes A_2) \end{array} $$

is commutative.

*Remark 2.7.* For $i = 1, 2$ fix $S_i \in \mathcal{S}$. In view of axiom (S-fib-0) of Definition 1.1, condition (mETS) in the case where $f_i = g_i = \mathrm{id}_{S_i}$ means that the natural isomorphism of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ m_{\mathrm{id}_{S_1}, \mathrm{id}_{S_2}} : A_1 \boxtimes A_2 \xrightarrow{\sim} A_1 \boxtimes A_2 $$

is idempotent. Hence it must be the identity.

As we now explain, the result of Lemma 2.4 has an analogue in the setting of external tensor structures, which clarifies the meaning of axiom (mETS): between any two given iterations of external tensor product functors and inverse image functors (respecting the single factors), it is possible to construct at most one natural isomorphism by combining connection isomorphisms and external monoidality isomorphisms in any possible order.

**Lemma 2.8.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$ and, for $i = 1, 2$, let $f_i : T_i \to S_i$ be a morphism in $\mathcal{S}$. For every choice of two composable pairs of composable tuples in $\mathcal{S}$

$$ \underline{g}_i = (g_{i,1}, \dots, g_{i,r}), \quad \underline{h}_i = (h_{i,1}, \dots, h_{i,s}) \quad (i=1,2) $$

satisfying $h_{i,s} \circ \dots \circ h_{i,1} \circ g_{i,r} \circ \dots \circ g_{i,1} = f_i$, consider the functor $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(T_1 \times T_2)$

$$ F^{\boxtimes}_{(g_1^1, h_1^1), (g_2^1, h_2^1)}(A_1, A_2) := (g_1^1 \times g_2^1)^* (h_1^* A_1 \boxtimes h_2^* A_2). $$

Then, given, for $i = 1, 2$, any two pairs of tuples $(g_i^{(1)}, h_i^{(1)})$ and $(g_i^{(2)}, h_i^{(2)})$ as above, all isomorphisms of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(T_1 \times T_2)$

$$ F^{\boxtimes}_{(g_1^{(1)}, h_1^{(1)}), (g_2^{(1)}, h_2^{(1)})} \xrightarrow{\sim} F^{\boxtimes}_{(g_1^{(2)}, h_1^{(2)}), (g_2^{(2)}, h_2^{(2)})} $$

obtained by composing connection isomorphisms and external monoidality isomorphisms (and inverses thereof) coincide.
As in the case of internal tensor structures, we can reduce the proof of this result to the setting of usual
fibered categories considered in Section 1 via an auxiliary construction:

**Construction 2.9.** Let $(\boxtimes, m)$ be an internal tensor structure on $\mathbb{H}$. Then, following Notation 1.9, we can construct an $(\mathcal{S}^2 \times \mathcal{I})$-fibered category $\mathbb{H}^{\boxtimes}$ out of it, as follows:

* For every $S_1, S_2 \in \mathcal{S}$ we set $\mathbb{H}^{\boxtimes}(S_1, S_2; 1) := \mathbb{H}(S_1) \times \mathbb{H}(S_2)$ and $\mathbb{H}^{\boxtimes}(S_1, S_2; 2) := \mathbb{H}(S_1 \times S_2)$.

* Given a morphism $\phi$ in $\mathcal{S}^2 \times \mathcal{I}$, we define the functor $\phi^*$ according to the type of $\phi$:

- if $\phi = (f_1, f_2; 1) : (T_1, T_2; 1) \to (S_1, S_2; 1)$, we set $\phi^*(A_1, A_2) := (f_1^* A_1, f_2^* A_2)$;

- if $\phi = (f, 2) : (T_1, T_2; 2) \to (S_1, S_2; 2)$, we set $\phi^* A := (f_1 \times f_2)^* A$;

- if $\phi = r_{S_1, S_2} \circ (f_1, f_2; 2)$, we set $\phi^*(A_1, A_2) := (f_1 \times f_2)^*(A_1 \boxtimes A_2)$.

* Given two composable morphisms $\phi$ and $\psi$ in $\mathcal{S}^2 \times \mathcal{I}$, we define the connection isomorphism $\text{conn}_{\phi, \psi}$ according to the type of $\phi$ and $\psi$:

(i) if $\phi = (f_1, f_2; 1)$ and $\psi = (g_1, g_2; 1)$, we set

$$ \text{conn}_{\phi, \psi}(A_1, A_2) : ((g_1 f_1)^* A_1, (g_2 f_2)^* A_2) \xrightarrow{(\text{conn}_{f_1, g_1}, \text{conn}_{f_2, g_2})} (f_1^* g_1^* A_1, f_2^* g_2^* A_2); $$

(ii) if $\phi = (f_1, f_2; 2)$ and $\psi = (g_1, g_2; 2)$, we set

$$ \text{conn}_{\phi, \psi}(A) : (g_1 f_1 \times g_2 f_2)^* A \xrightarrow{\text{conn}_{f_1 \times f_2, g_1 \times g_2}} (f_1 \times f_2)^* (g_1 \times g_2)^* A; $$

(iii) if $\phi = (f, 2)$ and $\psi = r_V \circ (g, 2)$, we set

$$ \text{conn}_{\phi, \psi}(A_1, A_2) : (g_1 f_1 \times g_2 f_2)^* (A_1 \boxtimes A_2) \xrightarrow{\text{conn}_{f_1 \times f_2, g_1 \times g_2}} (f_1 \times f_2)^* (g_1 \times g_2)^* (A_1 \boxtimes A_2); $$

(iv) if $\phi = r_S \circ (f, 2)$ and $\psi = (g, 1)$, we set

$$ \text{conn}_{\phi, \psi}(A_1, A_2) : \begin{array}{r@{\ }c@{\ }l@{\quad}l@{\quad}l} & (g_1 f_1 \times g_2 f_2)^* (A_1 \boxtimes A_2) & \arrow{->} & \text{conn}_{f_1 \times f_2, g_1 \times g_2} \\ & (f_1 \times f_2)^* (g_1 \times g_2)^* (A_1 \boxtimes A_2) & \arrow{->} & (f_1 \times f_2)^* (g_1^* A_1 \boxtimes g_2^* A_2). \end{array} $$

Arguing in the same way as in Construction 2.5, one can check that this construction indeed defines a $(\mathcal{S}^2 \times \mathcal{I})$-fibered category; we omit the details.

*Proof of Lemma 2.8.* As a consequence of Construction 2.9, this is just a particular case of Lemma 1.3. □

**2.2. Correspondence between internal and external tensor structures.** We now move towards the main goal of the present section: we want to prove that, modulo equivalence, giving an internal tensor structure on $\mathbb{H}$ is the same as giving an external tensor structure on it.

The following result shows how to extend the classical formula (2) to a construction from internal to external tensor structures:

**Lemma 2.10.** Let $(\boxtimes, m)$ be an internal tensor structure on $\mathbb{H}$. Then, assigning

* to every $S_1, S_2 \in \mathcal{S}$, the functor

$$ - \boxtimes - = - \boxtimes_{S_1, S_2} : \mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2) $$

defined by the formula

$$ A_1 \boxtimes A_1 := \text{pr}_1^{12*} A_1 \otimes \text{pr}_2^{12*} A_2, $$

* to every choice of morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, $i = 1, 2$, the natural isomorphism of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(T_1 \times T_2)$

$$ m^e = m^e_{f_1, f_2} : f_1^* A_1 \boxtimes f_2^* A_2 \xrightarrow{\sim} (f_1 \times f_2)^* (A_1 \boxtimes A_2) $$
defined as the composite

$$ \begin{array}{ccc} & f_1^* A_1 \boxtimes f_2^* A_2 & (f_1 \times f_2)^*(A_1 \boxtimes A_2) \\ \parallel & & \parallel \\ pr_1^{12,*} f_1^* A_1 \otimes pr_2^{12,*} f_2^* A_2 = & (f_1 \times f_2)^* pr_1^{12,*} A_1 \otimes (f_1 \times f_2)^* pr_2^{12,*} A_2 \xrightarrow{m} (f_1 \times f_2)^* (pr_1^{12,*} A_1 \otimes pr_2^{12,*} A_2) \end{array} $$

defines an external tensor structure (⊗, m^e) on H.

*Proof.* Let us check that this definition satisfies condition (mETS): given, for $i = 1, 2$, two composable morphisms $f_i : T_i \to S_i$ and $g_i : S_i \to V_i$ in $\mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(V_1) \times \mathbb{H}(V_2) \to \mathbb{H}(T_1 \times T_2)$

$$ \begin{array}{ccc} & (g_1 f_1)^* A_1 \boxtimes (g_2 f_2)^* A_2 & m^e \\ \parallel & & \parallel \\ f_1^* g_1^* A_1 \boxtimes f_2^* g_2^* A_2 \xrightarrow{m^e} (f_1 \times f_2)^* (g_1^* A_1 \boxtimes g_2^* A_2) \xrightarrow{m^e} (f_1 \times f_2)^* (g_1 \times g_2)^* (A_1 \boxtimes A_2) \end{array} $$

is commutative. Unwinding the definitions in terms of the internal tensor product, we obtain the more explicit diagram

$$ \begin{array}{ccc} & (g_{12} f_{12})^* pr_1^{12,*} A_1 \otimes (g_{12} f_{12})^* pr_2^{12,*} A_2 & m \\ \parallel & & \parallel \\ pr_1^{12,*} (g_1 f_1)^* A_1 \otimes pr_2^{12,*} (g_2 f_2)^* A_2 & f_{12}^* g_{12}^* (pr_1^{12,*} A_1 \otimes pr_2^{12,*} A_2) \\ \parallel & & \parallel \\ pr_1^{12,*} f_1^* g_1^* A_1 \otimes pr_2^{12,*} f_2^* g_2^* A_2 & f_{12}^* (g_{12}^* pr_1^{12,*} A_1 \otimes g_{12}^* pr_2^{12,*} A_2) \\ \parallel & & \parallel \\ f_{12}^* pr_1^{12,*} g_1^* A_1 \otimes f_{12}^* pr_2^{12,*} g_2^* A_2 & m \\ \parallel & & \parallel \\ f_{12}^* g_{12}^* pr_1^{12,*} A_1 \otimes f_{12}^* g_{12}^* pr_2^{12,*} A_2 & m \\ \end{array} $$

that we decompose as

<figure><img src="image_4.png" /></figure>

Here, the left-most piece is commutative by Lemma 1.3, the upper-right piece is commutative by axiom (mITS), and the lower-right piece is commutative by naturality. This concludes the proof. □

Conversely, the following result shows how to turn the classical formula (3) into a construction from external to internal tensor structures:

**Lemma 2.11.** Let (⊗, m) be an external tensor structure on H. Then, assigning
Proof. Start from an internal tensor structure $(\otimes, m)$ on $\mathbb{H}$, and let $(\otimes', m')$ denote the internal tensor structure obtained from it by applying first Lemma 2.10 and then Lemma 2.11: concretely, for every $S \in \mathcal{S}$, it is defined by the formula

$$A \otimes' B := \Delta_S^*(pr_1^{12,*} A \otimes pr_2^{12,*} B).$$

We want to construct a canonical equivalence $e: (\otimes, m) \xrightarrow{\sim} (\otimes', m')$. To this end, for every $S \in \mathcal{S}$ we define an isomorphism of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$e = e_S : A \otimes B \xrightarrow{\sim} A \otimes' B$$

by taking the composite

$$A \otimes B = \Delta_S^* pr_1^{12,*} A \otimes \Delta_S^* pr_2^{12,*} B \xrightarrow{m} \Delta_S^* (pr_1^{12,*} A \otimes pr_2^{12,*} B) =: A \otimes' B.$$

We claim that, as $S$ varies in $\mathcal{S}$, these isomorphisms satisfy condition (eq-ITS): given a morphism $f: T \to S$ in $\mathcal{S}$, we have to check that the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$\begin{tikzcd}
    f^* A \otimes f^* B \arrow[r, "m"] \arrow[d, "e"] & f^*(A \otimes B) \arrow[d, "e"] \\
    f^* A \otimes' f^* B \arrow[r, "m'"'] & f^*(A \otimes' B)
\end{tikzcd}$$

is commutative. Unwinding the definitions, we obtain the more explicit diagram

$$\begin{tikzcd}
    \Delta_T^* pr_1^{12,*} f^* A \otimes \Delta_T^* pr_2^{12,*} f^* B \arrow[r, "m"] & \Delta_T^* (pr_1^{12,*} f^* A \otimes pr_2^{12,*} f^* B) \arrow[d, "\|"'] \\
    f^* A \otimes f^* B \arrow[r, "m"] & \Delta_T^* ((f \times f)^* pr_1^{12,*} A \otimes (f \times f)^* pr_2^{12,*} B) \arrow[d, "m"'] \\
    f^*(A \otimes B) & \Delta_T^* (f \times f)^* (pr_1^{12,*} A \otimes pr_2^{12,*} B) \arrow[d, "\|"']
\end{tikzcd}$$

that we decompose as

$$\begin{tikzcd}
    \bullet \arrow[r, "m"] \arrow[d, "\bullet"] & \bullet \arrow[r, "m"] \arrow[d, "\bullet"] \\
    \bullet \arrow[r, "m"] \arrow[d, "\bullet"] & \bullet \arrow[r, "m"] \arrow[d, "\bullet"]
\end{tikzcd}$$

Here, the upper-left piece is commutative by Lemma 1.3, the lower-left and upper-right pieces are commutative by naturality, and the lower-right piece is commutative by Lemma 2.4. This proves the claim.

Conversely, start from an external tensor structure $(\boxtimes, m)$ on $\mathbb{H}$, and let $(\boxtimes', m')$ denote the external tensor structure obtained from it by applying first Lemma 2.11 and then Lemma 2.10: concretely, for every $S_1, S_2 \in \mathcal{S}$, it is defined by the formula

$$A_1 \boxtimes' A_2 := \Delta_{S_1 \times S_2}^* (pr_1^* A_1 \boxtimes pr_2^* A_2).$$
additional constraints, since we do not need such an extension for our applications; however, we often make use of these two auxiliary results in order to simplify our computations.

### 3. ASSOCIATIVITY CONSTRAINTS

The goal of this section is to introduce associativity constraint in the setting of internal and of external tensor structures and to compare them.

The notion of associativity constraint for single monoidal categories comes from [Mac63, § 3]; in the case of fibered categories, one needs to require inverse image functors to respect associativity constraints as in [Ayo07a, Defn. 2.1.85(1)]. The latter definition can be formulated in our language of internal tensor structures as follows:

**Definition 3.1.** Let $(\otimes, m)$ be an internal tensor structure on $\mathbb{H}$.

(1) An *internal associativity constraint* $a$ on $(\otimes, m)$ is the datum of

* for every $S \in \mathcal{S}$, a natural isomorphism between functors $\mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$a = a_S : (A \otimes B) \otimes C \xrightarrow{\sim} A \otimes (B \otimes C)$$

satisfying the following conditions:

(aITS-1) For every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$((A \otimes B) \otimes C) \otimes D \xrightarrow{\substack{a \\ \downarrow^a}} (A \otimes B) \otimes (C \otimes D) \\ (A \otimes (B \otimes C)) \otimes D \xrightarrow{\substack{a \\ \downarrow^a}} A \otimes ((B \otimes C) \otimes D) \xrightarrow{\substack{a \\ \downarrow^a}} A \otimes (B \otimes (C \otimes D))$$

is commutative.

(aITS-2) For every morphism $f: T \to S$ in $\mathcal{S}$, the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$(f^* A \otimes f^* B) \otimes f^* C \xrightarrow{\substack{m \\ \downarrow^a}} f^*(A \otimes B) \otimes f^* C \xrightarrow{\substack{m \\ \downarrow^a}} f^*((A \otimes B) \otimes C) \\ f^* A \otimes (f^* B \otimes f^* C) \xrightarrow{\substack{m \\ \downarrow^a}} f^* A \otimes f^*(B \otimes C) \xrightarrow{\substack{m \\ \downarrow^a}} f^*(A \otimes (B \otimes C))$$

is commutative.

We say that $(\otimes, m; a)$ is an *associative internal tensor structure* on $\mathbb{H}$.

(2) Let $(\otimes, m; a)$ and $(\otimes', m'; a')$ be two associative internal tensor structures on $\mathbb{H}$. An *equivalence of associative internal tensor structures* $e: (\otimes, m; a) \rightsquigarrow (\otimes', m; a')$ is an equivalence of internal tensor structures $e: (\otimes, m) \rightsquigarrow (\otimes', m')$ satisfying the following additional condition:

(eq-aITS) For every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$(A \otimes B) \otimes C \xrightarrow{\substack{e \\ \downarrow^a}} (A \otimes' B) \otimes C \xrightarrow{\substack{e \\ \downarrow^a}} (A \otimes' B) \otimes' C \\ A \otimes (B \otimes C) \xrightarrow{\substack{e \\ \downarrow^a}} A \otimes (B \otimes' C) \xrightarrow{\substack{e \\ \downarrow^a}} A \otimes' (B \otimes' C)$$

is commutative.

In order to get a similar notion in the setting of external tensor structures, we just need to transliterate the previous conditions in the obvious way, as follows:

**Definition 3.2.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$.

(1) An *external associativity constraint* $a$ on $(\boxtimes, m)$ is the datum of
* for every $S_1, S_2, S_3 \in \mathcal{S}$, a natural isomorphism between functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(S_1 \times S_2 \times S_3)$

$$a = a_{S_1,S_2,S_3} : (A_1 \boxtimes A_2) \boxtimes A_3 \xrightarrow{\sim} A_1 \boxtimes (A_2 \boxtimes A_3)$$

satisfying the following conditions:

(aETS-1) For every $S_1, S_2, S_3, S_4 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \times \mathbb{H}(S_4) \to \mathbb{H}(S_1 \times S_2 \times S_3 \times S_4)$

$$((A_1 \boxtimes A_2) \boxtimes A_3) \boxtimes A_4 \xrightarrow{\substack{a \\ \downarrow a}} (A_1 \boxtimes A_2) \boxtimes (A_3 \boxtimes A_4) \\ (A_1 \boxtimes (A_2 \boxtimes A_3)) \boxtimes A_4 \xrightarrow{\substack{a \\ \downarrow a}} A_1 \boxtimes ((A_2 \boxtimes A_3) \boxtimes A_4) \xrightarrow{\substack{a \\ \downarrow a}} A_1 \boxtimes (A_2 \boxtimes (A_3 \boxtimes A_4))$$

is commutative.

(aETS-2) For every three morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, $i = 1, 2, 3$, the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(T_1 \times T_2 \times T_3)$

$$(f_1^* A_1 \boxtimes f_2^* A_2) \boxtimes f_3^* A_3 \xrightarrow{\substack{m \\ \downarrow a}} (f_1 \times f_2)^*(A_1 \boxtimes A_2) \boxtimes f_3^* A_3 \xrightarrow{\substack{m \\ \downarrow a}} (f_1 \times f_2 \times f_3)^*((A_1 \boxtimes A_2) \boxtimes A_3) \\ f_1^* A_1 \boxtimes (f_2^* A_2 \boxtimes f_3^* A_3) \xrightarrow{\substack{m \\ \downarrow a}} f_1^* A_1 \boxtimes (f_2 \times f_3)^*(A_2 \boxtimes A_3) \xrightarrow{\substack{m \\ \downarrow a}} (f_1 \times f_2 \times f_3)^*(A_1 \boxtimes (A_2 \boxtimes A_3))$$

is commutative.

We say that $(\boxtimes, m; a)$ is an associative external tensor structure on $\mathbb{H}$.

(2) Let $(\boxtimes', m'; a')$ and $(\boxtimes, m; a)$ be two associative external tensor structures on $\mathbb{H}$. An equivalence of associative external tensor structures $e : (\boxtimes', m'; a') \xrightarrow{\sim} (\boxtimes, m; a)$ is an equivalence of external tensor structures $e : (\boxtimes', m') \xrightarrow{\sim} (\boxtimes, m)$ satisfying the following additional condition:

(eq-aETS) For every $S_1, S_2, S_3 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(S_1 \times S_2 \times S_3)$

$$(A_1 \boxtimes' A_2) \boxtimes' A_3 \xrightarrow{e} (A_1 \boxtimes A_2) \boxtimes' A_3 \xrightarrow{e} (A_1 \boxtimes A_2) \boxtimes A_3 \\ A_1 \boxtimes' (A_2 \boxtimes' A_3) \xrightarrow{e} A_1 \boxtimes' (A_2 \boxtimes A_3) \xrightarrow{e} A_1 \boxtimes (A_2 \boxtimes A_3)$$

is commutative.

As promised, we now describe how associativity constraints can be translated between the internal and the external setting along the constructions of Lemma 2.10 and Lemma 2.11. This is the content of the following two results:

**Lemma 3.3.** Let $(\boxtimes, m)$ be an internal tensor structure on $\mathbb{H}$, and let $(\boxtimes, m^e)$ denote the external tensor structure defined from it via Lemma 2.10. Suppose that we are given an internal associativity constraint $a$ on $(\boxtimes, m)$. Then, associating

* to every $S_1, S_2, S_3 \in \mathcal{S}$, the natural isomorphism of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(S_1 \times S_2 \times S_3)$

$$a^e = a_{S_1,S_2,S_3}^e : (A_1 \boxtimes A_2) \boxtimes A_3 \xrightarrow{\sim} A_1 \boxtimes (A_2 \boxtimes A_3)$$
defined by taking the composite

<figure><img src="image_5.png" /></figure>

defines an external associativity constraint $a^e$ on $(\boxtimes, m^e)$.

*Proof.* We have to check that the definition satisfies conditions (aETS-1) and (aETS-2).

We start from condition (aETS-1): given $S_1, S_2, S_3, S_4 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \times \mathbb{H}(S_4) \to \mathbb{H}(S_1 \times S_2 \times S_3 \times S_4)$

<figure><img src="image_6.png" /></figure>

is commutative. To this end, we decompose the diagram as

<figure><img src="image_7.png" /></figure>

where, for notational convenience, we have set $B_i := pr_i^{1234,*} A_i$, $i = 1,2,3,4$. The five unnamed natural isomorphisms are obtained by combining connection isomorphisms and internal monoidality isomorphisms along suitable paths; by Lemma 2.4, we know that the definitions do not depend on the chosen paths. Since the central rectangle in the latter diagram is commutative by axiom (aITS-1), it now suffices to show that the five lateral pieces are commutative as well. For each of them, it suffices to expand the definition of the two auxiliary isomorphisms involved in the most convenient way, using again the independence result of Lemma 2.4; we omit the details.

We now treat condition (aETS-2): given three morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, $i = 1,2,3$, we have to show that the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(T_1 \times T_2 \times T_3)$

<figure><img src="image_8.png" /></figure>
is commutative. To this end, we decompose the diagram as

<figure><img src="image_9.png" /></figure>

where, for notational convenience, we have set $B_i := \text{pr}_i^{123,*} A_i$, $i = 1, 2, 3$. The six unnamed natural isomorphisms are obtained by combining connection isomorphisms and internal monoidity isomorphisms along suitable paths; by Lemma 2.4, we know that the definitions do not depend on the chosen paths. Since the central rectangle is the latter diagram is commutative by axiom (aITS-2), it now suffices to show that the six lateral pieces are commutative as well. This can be achieved using the same trick as for the first part of the proof; we omit the details. □

**Lemma 3.4.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$, and let $(\otimes, m^i)$ denote the internal tensor structure defined from it via Lemma 2.11. Suppose that we are given an external associativity constraint $a$ on $(\boxtimes, m)$. Then, associating

* to every $S \in \mathcal{S}$, the natural isomorphism of functors $\mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$a^i = a_S^i : (A \otimes B) \otimes C \xrightarrow{\sim} A \otimes (B \otimes C)$$

defined by taking the composite

$$ \begin{array}{ccc} (A \otimes B) \otimes C & & A \otimes (B \otimes C) \\ \parallel & & \parallel \\ \Delta_S^*(\Delta_S^*(A \boxtimes B) \boxtimes C) & & \Delta_S^*(A \boxtimes \Delta_S^*(B \boxtimes C)) \\ \downarrow^m & & \downarrow^m \\ \Delta_S^*(\Delta_S \times \text{id}_S)^*((A \boxtimes B) \boxtimes C) & & \Delta_S^*(\text{id}_S \times \Delta_S)^*((A \boxtimes (B \boxtimes C)) \\ \parallel & & \parallel \\ \Delta_S^{(3),*}((A \boxtimes B) \boxtimes C) & \xrightarrow{a} & \Delta_S^{(3),*}(A \boxtimes (B \boxtimes C)) \end{array} $$

defines an internal associativity constraint $a^i$ on $(\otimes, m^i)$.

*Proof.* We have to check that the definition satisfies conditions (aITS-1) and (aITS-2).

We start from condition (aITS-1): given $S \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{array}{ccc} ((A \otimes B) \otimes C) \otimes D & \xrightarrow{a^i} & (A \otimes B) \otimes (C \otimes D) \\ \downarrow^{a^i} & & \downarrow^{a^i} \\ (A \otimes (B \otimes C)) \otimes D & \xrightarrow{a^i} & A \otimes ((B \otimes C) \otimes D) \xrightarrow{a^i} A \otimes (B \otimes (C \otimes D)) \end{array} $$
*Proof.* We keep the same notation adopted in the proof of Proposition 2.12.

Start from an associative internal tensor structure $(\otimes, m; a)$ on $\mathbb{H}$, and let $a'$ denote the internal associativity constraint on $(\otimes', m')$ obtained from $a$ by applying first Lemma 3.3 and then Lemma 3.4. We claim that the equivalence of internal tensor structures $e : (\otimes, m) \xrightarrow{\sim} (\otimes', m')$ constructed in the first part of the proof of Proposition 2.12 satisfies condition (eq-aITS): given $S \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{array}{ccc} (A \otimes B) \otimes C & \xrightarrow{e} & (A \otimes' B) \otimes C \xrightarrow{e} (A \otimes' B) \otimes' C \\ \bigdownarrow^{a} & & \bigdownarrow^{a'} \\ A \otimes (B \otimes C) & \xrightarrow{e} & A \otimes (B \otimes' C) \xrightarrow{e} A \otimes' (B \otimes' C) \end{array} $$

is commutative. Expanding the definitions, we find that the internal associativity isomorphism $a'$ is given by the composite

$$ \begin{array}{ccc} (A \otimes' B) \otimes' C & \xrightarrow{\parallel} & (A \otimes' (B \otimes' C)) \\ \Delta_S^* (pr_1^{12,*} \Delta_S^* (pr_1^{12,*} A \otimes pr_2^{12,*} B) \otimes pr_2^{12,*} C) & \xrightarrow{\parallel} & \Delta_S^* (pr_1^{12,*} A \otimes pr_2^{12,*} \Delta_S^* (pr_1^{12,*} B \otimes pr_2^{12,*} C)) \\ \Delta_S^* ((\Delta_S \times \mathrm{id}_S)^* pr_{12}^{12,*} (pr_1^{12,*} A \otimes pr_2^{12,*} B) \otimes (\Delta_S \times \mathrm{id}_S)^* pr_3^{123,*} C) & \xrightarrow{\parallel} & \Delta_S^* ((\Delta_S \times \mathrm{id}_S)^* pr_1^{123,*} A \otimes (\Delta_S \times \mathrm{id}_S)^* pr_{23}^{123,*} (pr_2^{23,*} B \otimes pr_3^{23,*} C)) \\ \bigdownarrow^{m} & & \bigdownarrow^{m} \\ \Delta_S^* (\Delta_S \times \mathrm{id}_S)^* (pr_{12}^{123,*} (pr_1^{12,*} A \otimes pr_2^{12,*} B) \otimes pr_3^{123,*} C) & \xrightarrow{\parallel} & \Delta_S^* (\mathrm{id}_S \times \Delta_S)^* (pr_1^{123,*} A \otimes pr_{23}^{123,*} (pr_2^{23,*} B \otimes pr_3^{23,*} C)) \\ \Delta_S^{(3),*} (pr_{12}^{123,*} (pr_1^{12,*} A \otimes pr_2^{12,*} B) \otimes pr_3^{123,*} C) & \xrightarrow{\parallel} & \Delta_S^{(3),*} (pr_1^{123,*} A \otimes pr_{23}^{123,*} (pr_2^{23,*} B \otimes pr_3^{23,*} C)) \\ \biguparrow^{m} & & \biguparrow^{m} \\ \Delta_S^{(3),*} ((pr_{12}^{123,*} pr_1^{12,*} A \otimes pr_{12}^{123,*} pr_2^{12,*} B) \otimes pr_3^{123,*} C) & \xrightarrow{\parallel} & \Delta_S^{(3),*} (pr_1^{123,*} A \otimes (pr_{23}^{123,*} pr_2^{23,*} B \otimes pr_{23}^{123,*} pr_3^{23,*} C)) \\ \Delta_S^{(3),*} ((pr_1^{123,*} A \otimes pr_2^{123,*} B) \otimes pr_3^{123,*} C) & \xrightarrow{\substack{a \\ \parallel}} & \Delta_S^{(3),*} (pr_1^{123,*} A \otimes (pr_2^{123,*} B \otimes pr_3^{123,*} C)) \end{array} $$

Substituting this composite in the above diagram, we obtain a diagram of the form

$$ \begin{array}{ccc} (A \otimes B) \otimes C & \xrightarrow{\sim} & \Delta_S^{(3),*}((pr_1^{123,*} A \otimes pr_2^{123,*} B) \otimes pr_3^{123,*} C) \\ \bigdownarrow^{a} & & \bigdownarrow^{a} \\ A \otimes (B \otimes C) & \xrightarrow{\sim} & \Delta_S^{(3),*} (pr_1^{123,*} A \otimes (pr_2^{123,*} B \otimes pr_3^{123,*} C)) \end{array} $$

where the two horizontal arrows are composites of connection isomorphisms and internal monoidality isomorphisms along suitable paths. Since, by Lemma 2.4, we know that the resulting isomorphisms are independent of the chosen paths, we deduce that the latter diagram coincides, for example, with the outer part
of the diagram

<figure><img src="image_10.png" /></figure>

where the left-most piece is commutative by naturality while the right-most piece is commutative by axiom (aITS-2). This proves the claim.

Conversely, start from an associative external tensor structure (⊗, m; a) on H, and let a' denote the external associativity constraint on (⊗', m') obtained from a by applying first Lemma 3.4 and then Lemma 3.3. We claim that the equivalence of external tensor structures e : (⊗', m') $\xrightarrow{\sim}$ (⊗, m) constructed in the second part of the proof of Proposition 2.12 satisfies condition (eq-aETS): given S₁, S₂, S₃ ∈ ℹ, we have to show that the diagram of functors H(S₁) × H(S₂) × H(S₃) → H(S₁ × S₂ × S₃)

<figure><img src="image_11.png" /></figure>

is commutative. Expanding the definitions, we find that the external associativity isomorphism a' is given by the composite

<figure><img src="image_12.png" /></figure>
Substituting this composite in the above diagram, we obtain a diagram of the form

$$
\begin{align*}
\Delta_{S_{123}}^{(3),*}((pr_1^{123,*} A_1 \boxtimes pr_2^{123,*} A_2) \boxtimes pr_3^{123,*} A_3) &\xrightarrow{\sim} (A_1 \boxtimes A_2) \boxtimes A_3 \\
\Delta_{S_{123}}^{(3),*}(pr_1^{123,*} A_1 \boxtimes (pr_2^{123,*} A_2 \boxtimes pr_3^{123,*} A_3)) &\xrightarrow{\sim} A_1 \boxtimes (A_2 \boxtimes A_3)
\end{align*}
$$

where the two horizontal arrows are composites of connection isomorphisms and internal monoidality iso-
morphisms along suitable paths. Since, by Lemma 2.8, we know that the resulting isomorphisms are inde-
pendent of the chosen paths, we deduce that the latter diagram coincides, for example, with the outer part
of the diagram

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
\Delta_{S_{123}}^{(3),*}((pr_1^{123} \times pr_2^{123})^* (A_1 \boxtimes A_2) \boxtimes pr_3^{123,*} A_3) \arrow[r, "m"] & \Delta_{S_{123}}^{(3),*}(pr_1^{123} \times pr_2^{123} \times pr_3^{123})^* ((A_1 \boxtimes A_2) \boxtimes A_3) \arrow[r, "="{yshift=0.5ex}", "(A_1 \boxtimes A_2) \boxtimes A_3"] & (A_1 \boxtimes A_2) \boxtimes A_3 \\
\Delta_{S_{123}}^{(3),*}((pr_1^{123,*} A_1 \boxtimes pr_2^{123,*} A_2) \boxtimes pr_3^{123,*} A_3) \arrow[r, "m"] & \\
\Delta_{S_{123}}^{(3),*}(pr_1^{123,*} A_1 \boxtimes (pr_2^{123,*} A_2 \boxtimes pr_3^{123,*} A_3)) \arrow[r, "m"] & \Delta_{S_{123}}^{(3),*}(pr_1^{123} \times pr_2^{123} \times pr_3^{123})^* (A_1 \boxtimes (A_2 \boxtimes A_3)) \arrow[r, "="{yshift=0.5ex}", "(A_1 \boxtimes (A_2 \boxtimes A_3))"] & A_1 \boxtimes (A_2 \boxtimes A_3)
\end{tikzcd}
$$

where the left-most piece is commutative by axiom (aETS-2) while the right-most piece is commutative by
naturality. This proves the claim and concludes the proof. $\square$

4. COMMUTATIVITY CONSTRAINTS

The goal of this section is to introduce commutativity constraints in the setting of internal and external tensor structures and to compare them.

The notion of commutativity constraint for single monoidal categories comes from [Mac63, § 4]; in the case of fibered categories, one needs to require inverse image functors to respect commutativity constraints as in [Ayo07a, Defn. 2.1.86]. The latter definition can be formulated in the language of internal tensor structures as follows:

**Definition 4.1.** Let $(\otimes, m)$ be an internal tensor structure on $\mathbb{H}$.

(1) An internal commutativity constraint $c$ on $(\otimes, m)$ is the datum of
* for every $S \in \mathcal{S}$, a natural isomorphism between functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$
c = c_S : A \otimes B \xrightarrow{\sim} B \otimes A
$$

satisfying the following condition

(cITS-1) For every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
A \otimes B \arrow[r, "c"] & B \otimes A \\
\arrow[d, "\text{id}", "c"] & \\
A \otimes B \arrow[r, "c"] &
\end{tikzcd}
$$

is commutative.
(cITS-2) For every morphism $f: T \to S$ in $\mathcal{S}$, the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ \begin{array}{ccc} f^*A \otimes f^*B & \xrightarrow{c} & f^*B \otimes f^*A \\ \bigdownarrow^m & & \bigdownarrow^m \\ f^*(A \otimes B) & \xrightarrow{c} & f^*(B \otimes A) \end{array} $$

is commutative.

We say that $(\otimes, m; c)$ is a symmetric internal tensor structure on $\mathbb{H}$.

(2) Let $(\otimes, m; c)$ and $(\otimes', m'; c')$ be two symmetric internal tensor structures on $\mathbb{H}$. An equivalence of symmetric internal tensor structures $e: (\otimes, m; c) \xrightarrow{\sim} (\otimes', m'; c')$ is an equivalence of internal tensor structures $e: (\otimes, m) \xrightarrow{\sim} (\otimes', m')$ satisfying the following additional condition:

(eq-cITS) For every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{array}{ccc} A \otimes B & \xrightarrow{e} & A \otimes' B \\ \bigdownarrow^c & & \bigdownarrow^c \\ B \otimes A & \xrightarrow{e} & B \otimes' A \end{array} $$

is commutative.

In order to get a similar notion in the setting of external tensor structures, we need to reformulate the operation of switching factors with the help of suitable permutation isomorphisms, as follows:

**Definition 4.2.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$.

(1) An external commutativity constraint $c$ on $(\boxtimes, m)$ is the datum of

* for every $S_1, S_2 \in \mathcal{S}$, a natural isomorphism between functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ c = c_{S_1, S_2} : A_1 \boxtimes A_2 \xrightarrow{\sim} \tau^*(A_2 \boxtimes A_1), $$

where $\tau = \tau_{(12)}: S_1 \times S_2 \to S_2 \times S_1$ denotes the permutation isomorphism satisfying the following conditions:

(cETS-1) For every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{array}{ccc} A_1 \boxtimes A_2 & \xrightarrow{c} & \tau^*(A_2 \boxtimes A_1) \\ \multicolumn{3}{c}{\diagdown} \\ \multicolumn{3}{c}{\tau^*\tau^*(A_1 \boxtimes A_2)} \end{array} $$

is commutative.

(cETS-2) For every choice of of morphisms $f_i: T_i \to S_i$ in $\mathcal{S}$, $i = 1, 2$, the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(T_2 \times T_1)$

$$ \begin{array}{ccc} f_1^* A_1 \boxtimes f_2^* A_2 & \xrightarrow{c} & \tau^*(f_2^* A_2 \boxtimes f_1^* A_1) \\ \bigdownarrow^m & & \bigdownarrow^m \\ (f_1 \times f_2)^*(A_1 \boxtimes A_2) & \xrightarrow{c} & (f_1 \times f_2)^* \tau^*(A_2 \boxtimes A_1) = \tau^*(f_2 \times f_1)^*(A_2 \boxtimes A_1) \end{array} $$

is commutative.

We say that $(\boxtimes, m; c)$ is a symmetric external tensor structure on $\mathbb{H}$.

(2) Let $(\boxtimes', m'; c')$ and $(\boxtimes, m; c)$ be two symmetric external tensor structures on $\mathbb{H}$. An equivalence of symmetric external tensor structures $e: (\boxtimes', m'; c') \xrightarrow{\sim} (\boxtimes, m; c)$ is an equivalence of external tensor structures $e: (\boxtimes', m') \xrightarrow{\sim} (\boxtimes, m)$ satisfying the following additional condition:
(eq-cETS) For every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{array}{ccc} A_1 \boxtimes' A_2 & \xrightarrow{e} & A_1 \boxtimes A_2 \\ \downarrow_{c'} & & \downarrow_c \\ \tau^*(A_2 \boxtimes' A_1) & \xrightarrow{e} & \tau^*(A_2 \boxtimes A_1) \end{array} $$

is commutative.

In the same spirit as for the previous section, we now describe how commutativity constraints can be translated between the internal and the external setting along the constructions of Lemma 2.10 and Lemma 2.11. This is the content of the following two results:

**Lemma 4.3.** Let $(\otimes, m)$ be an internal tensor structure on $\mathbb{H}$, and let $(\boxtimes, m^e)$ denote the external tensor structure defined from it via Lemma 2.10. Suppose that we are given an internal commutativity constraint $c$ on $(\otimes, m)$. Then, associating

* to every choice of $S_1, S_2 \in \mathcal{S}$, the natural isomorphism of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ (6) \qquad c^e = c_{S_1, S_2}^e : A_1 \boxtimes A_2 \xrightarrow{\sim} \tau^*(A_2 \boxtimes A_1) $$

defined by taking the composite

$$ \begin{array}{ccc} A_1 \boxtimes A_2 & & \tau^*(A_2 \boxtimes A_1) \\ \parallel & & \parallel \\ pr_1^{12,*} A_1 \otimes pr_2^{12,*} A_2 & & \tau^*(pr_2^{12,*} A_2 \otimes pr_1^{12,*} A_1) \\ \parallel & m \uparrow & \\ \tau^* pr_1^{12,*} A_1 \otimes \tau^* pr_2^{12,*} A_2 \xrightarrow{c} & \tau^* pr_2^{12,*} A_2 \otimes \tau^* pr_1^{12,*} A_1 & \end{array} $$

defines an external commutativity constraint $c^e$ on $(\boxtimes, m^e)$.

*Proof.* We have to show that the definition satisfies conditions (cETS-1) and (cETS-2).

We start from condition (cETS-1): given $S_1, S_2 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{array}{ccc} A_1 \boxtimes A_2 & \xrightarrow{c^e} & \tau^*(A_2 \boxtimes A_1) \\ \Big\downarrow & & \Big\downarrow \\ \tau^*\tau^*(A_1 \boxtimes A_2) & & \end{array} $$

is commutative. Expanding the definitions, we obtain the outer part of the diagram

$$ \begin{array}{ccc} pr_1^{12,*} A_1 \otimes pr_2^{12,*} A_2 & \xrightarrow{c} & \tau^* pr_1^{21,*} A_1 \otimes \tau^* pr_2^{21,*} A_2 \xrightarrow{m} & \tau^*(pr_2^{21,*} A_2 \otimes pr_1^{21,*} A_1) \\ & & \parallel & \\ \tau^*(\tau^* pr_2^{12,*} A_2 \otimes \tau^* pr_1^{12,*} A_1) & \downarrow_c & & \\ \tau^*(\tau^* pr_1^{12,*} A_1 \otimes \tau^* pr_2^{12,*} A_2) & \downarrow_m & & \\ \tau^*\tau^*(pr_1^{12,*} A_1 \otimes pr_2^{12,*} A_2) & & & \end{array} $$
that we decompose as

<figure><img src="image_13.png" /></figure>

Here, the upper central triangle is commutative by axiom (cITS-1) while the remaining pieces are com-
mutative by naturality, by axiom (cITS-2) and by Lemma 2.4. Hence the outer part of the diagram is
commutative as well.

We now treat condition (cETS-2): given two morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, $i = 1,2$, we have to show
that the diagram

$$
\begin{tikzcd}
& & f_1^* A_1 \boxtimes f_2^* A_2 \arrow[r, "c^e"] \arrow[d, "m^e"] & \tau^*(f_2^* A_2 \boxtimes f_1^* A_1) \arrow[d, "m^e"'] \\
(f_1 \times f_2)^*(A_1 \boxtimes A_2) \arrow[r, "c^e"] & (f_1 \times f_2)^* \tau^*(A_2 \boxtimes A_1) & &
\tau^*(f_2 \times f_1)^*(A_2 \boxtimes A_1)
\end{tikzcd}
$$

is commutative. Expanding the definitions, we obtain the diagram

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    & \tau^* pr_1^{21,*} f_1^* A_1 \otimes \tau^* pr_2^{21,*} f_2^* A_2 \arrow[r, "c"] & \tau^* pr_2^{21,*} f_2^* A_2 \otimes \tau^* pr_1^{21,*} f_1^* A_1 \arrow[r, "m"] & \tau^* (pr_2^{21,*} f_2^* A_2 \otimes pr_1^{21,*} f_1^* A_1) \arrow[d, "m"] \\
    & pr_1^{12,*} f_1^* A_1 \otimes pr_2^{12,*} f_2^* A_2 & & \\
    (f_1 \times f_2)^* pr_1^{12,*} A_1 \otimes (f_1 \times f_2)^* pr_2^{12,*} A_2 & & \\
    & (f_1 \times f_2)^* (pr_1^{12,*} A_1 \otimes pr_2^{12,*} A_2) & & \\
    & (f_1 \times f_2)^* (\tau^* pr_1^{21,*} A_1 \otimes \tau^* pr_2^{21,*} A_2) \arrow[r, "c"] \arrow[d, "m"] & (f_1 \times f_2)^* (\tau^* pr_2^{21,*} A_2 \otimes \tau^* pr_1^{21,*} A_1) \arrow[r, "m"] \arrow[d, "m"] & (f_1 \times f_2)^* \tau^* (pr_2^{21,*} A_2 \otimes pr_1^{21,*} A_1)
\end{tikzcd}
$$
that we decompose as

<figure><img src="image_14.png" /></figure>

Here, the lower-left piece is commutative by axiom (cITS-2) while the other pieces are commutative by naturality, by axiom (mITS) and by Lemma 1.3. Hence the outer part of the diagram is commutative as well. □

**Lemma 4.4.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$, and let $(\otimes, m^i)$ denote the external tensor structure defined from it via Lemma 2.11. Suppose that we are given an internal commutativity constraint $c$ on $(\boxtimes, m)$. Then, associating

* to every $S \in \mathcal{S}$, the natural isomorphism of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$c^i = c_S^i : A \otimes B \xrightarrow{\sim} B \otimes A$$

defined by taking the composite

$$\begin{tikzcd}
A \otimes B \arrow[r] \arrow[d] & B \otimes A \arrow[d] \\
\Delta_S^*(A \boxtimes B) \arrow[r, "c"] & \Delta_S^* \tau^*(B \boxtimes A) \arrow[r, "\cong"] & \Delta_S^*(B \boxtimes A)
\end{tikzcd}$$

defines an internal commutativity constraint $c^i$ on $(\boxtimes, m^i)$.

*Proof.* We have to show that the definition satisfies conditions (cITS-1) and (cITS-2).

We start from condition (cITS-1): given $S \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$\begin{tikzcd}
A \otimes B \arrow[r, "c"] \arrow[d, "\text{id}"] & B \otimes A \arrow[d] \\
A \otimes B \arrow[r, "c"] & A
\end{tikzcd}$$

is commutative. Expanding the definitions, we obtain the outer part of the diagram

$$\begin{tikzcd}
\Delta_S^*(A \boxtimes B) \arrow[r, "c"] & \Delta_S^* \tau^*(B \boxtimes A) \arrow[r, "\cong"] & \Delta_S^*(B \boxtimes A) \\
\phantom{\Delta_S^*} \arrow[d, "\text{id}"] & \phantom{\Delta_S^*} \arrow[d, "c"] & \phantom{\Delta_S^*} \\
\Delta_S^* \tau^*(A \boxtimes B) \arrow[r, "c"] & \Delta_S^* \tau^*(A \boxtimes B)
\end{tikzcd}$$
that we decompose as

<figure><img src="image_15.png" /></figure>

Here, the upper-right triangle is commutative by axiom (cETS-1) while the remaining pieces are commutative by naturality and by Lemma 1.3.

We now treat condition (cETS-2): given a morphism $f: T \to S$ in $\mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ \begin{array}{ccc} f^* A \otimes f^* B & \xrightarrow{c^i} & f^* B \otimes f^* A \\ \bigwedge^{m^i} & & \bigwedge^{m^i} \\ f^*(A \otimes B) & \xrightarrow{c^i} & f^*(B \otimes A) \end{array} $$

is commutative. Expanding the definitions, we obtain the diagram

$$ \begin{array}{ccc} \Delta_T^*(f^* A \boxtimes f^* B) & \xrightarrow{c} & \Delta_T^* \tau^*(f^* B \boxtimes f^* A) \\ \bigwedge^m & & \bigwedge^m \\ \Delta_T^*(f \times f)^*(A \boxtimes B) & & \Delta_T^*(f \times f)^*(B \boxtimes A) \\ \Vert & & \Vert \\ f^* \Delta_S^*(A \boxtimes B) & \xrightarrow{c} & f^* \Delta_S^* \tau^*(B \boxtimes A) \\ \Vert & & \Vert \\ \end{array} $$

that we decompose as

<figure><img src="image_16.png" /></figure>

Here, upper-left piece is commutative by axiom (cETS-2) while the other pieces are commutative by naturality and by Lemma 1.3. This proves the claim. □

**Proposition 4.5.** The constructions of Lemma 4.3 and Lemma 4.4 canonically define mutually inverse bijections between equivalence classes of symmetric internal and external tensor structures on $\mathbb{H}$.

*Proof.* We keep the same notation adopted in the proof of Proposition 2.12.

Start from a symmetric internal tensor structure $(\otimes, m; c)$ on $\mathbb{H}$, and let $c'$ denote the internal commutativity constraint on $(\otimes', m)$ obtained from $c$ by applying first Lemma 4.3 and then Lemma 4.4. We claim that the equivalence of internal tensor structures $e: (\otimes, m) \rightsquigarrow (\otimes', m')$ constructed in Proposition 2.12 satisfies
condition (eq-cITS): given $S \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{array}{ccc} A \otimes B & \xrightarrow{e} & A \otimes' B \\ \bigdownarrow^{c} & & \bigdownarrow^{c'} \\ B \otimes A & \xrightarrow{e} & B \otimes' A \end{array} $$

is commutative. Unwinding the definitions, we obtain the more explicit diagram

$$ \begin{array}{ccc} A \otimes B & \xrightarrow{c} & B \otimes A \\ \parallel & & \parallel \\ \Delta_S^* pr_1^{12,*} A \otimes \Delta_S^* pr_2^{12,*} B & \xrightarrow{m} & \Delta_S^* pr_2^{21,*} B \otimes \Delta_S^* pr_1^{21,*} A \\ \bigdownarrow^{m} & & \bigdownarrow^{m} \\ \Delta_S^*(pr_1^{12,*} A \otimes pr_2^{12,*} B) & \xrightarrow{c} & \Delta_S^*(pr_2^{21,*} B \otimes pr_1^{21,*} A) \\ \parallel & & \parallel \\ \Delta_S^*(\tau^* pr_1^{21,*} A \otimes \tau^* pr_2^{21,*} B) & \xrightarrow{c} & \Delta_S^*(\tau^* pr_2^{21,*} B \otimes \tau^* pr_1^{21,*} A) \xrightarrow{m} \Delta_S^* \tau^*(pr_2^{21,*} B \otimes pr_1^{21,*} A) \end{array} $$

that we decompose as

$$ \begin{array}{ccc} \bullet & & \bullet \\ \parallel & & \parallel \\ \bullet & c & \bullet \\ \parallel & & \parallel \\ \bullet & c & \bullet \\ \parallel & & \parallel \\ \bullet & c & \bullet \end{array} $$

Here, the left central piece is commutative by axiom (cITS-2) while the other pieces are commutative by axiom (mITS) and by naturality. This proves the claim.

Conversely, start from a symmetric external tensor structure $(\boxtimes, m; c)$ on $\mathbb{H}$, and let $c'$ denote the commutativity constraint on $(\boxtimes', m')$ obtained from $c$ by applying first Lemma 4.4 and then Lemma 4.3. We claim that the equivalence of external tensor structures $e: (\boxtimes', m') \rightsquigarrow (\boxtimes, m)$ constructed in the proof of Proposition 2.12 satisfies condition (eq-cETS): given $S_1, S_2 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{array}{ccc} A_1 \boxtimes' A_2 & \xrightarrow{e} & A_1 \boxtimes A_2 \\ \bigdownarrow^{c'} & & \bigdownarrow^{c} \\ \tau^*(A_2 \boxtimes' A_1) & \xrightarrow{e} & \tau^*(A_2 \boxtimes A_1) \end{array} $$
is commutative. Unwinding the definitions, we obtain the more explicit diagram

<figure><img src="image_17.png" /></figure>

that we decompose as

<figure><img src="image_18.png" /></figure>

Here, the central-left piece is commutative by axiom (cETS-2) while the other pieces are commutative by axiom (mETS) and by naturality. This proves the claim and concludes the proof. □

## 5. UNIT CONSTRAINTS

The goal of this section is to introduce unit constraints in the setting of internal and external tensor structures and to compare them; in particular, we introduce unit sections following the general notions fixed in Section 1.

The notion of unit constraint for a single monoidal category comes from [Mac63, § 5]; in the case of fibered categories, one needs to require inverse image functors to respect unit constraints as in [Ayo07a, Defn. 2.1.85(2)]. The latter definition can be formulated in the language of internal tensor structures as follows:

**Definition 5.1.** Let $(\otimes, m)$ be an internal tensor structure on $\mathbb{H}$.

(1) An internal unit constraint $u$ on $(\otimes, m)$ is the datum of

* a section $1$ of the $\mathcal{S}$-fibered category $\mathbb{H}$, called the unit section,

* for every $S \in \mathcal{S}$, two natural isomorphisms of functors $\mathbb{H}(S) \to \mathbb{H}(S)$

$$ u_r = u_{r,S} : A \otimes 1_S \xrightarrow{\sim} A, \qquad u_l = u_{l,S} : 1_S \otimes B \xrightarrow{\sim} B $$

satisfying the following conditions:
(uETS-1) For every choice of morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, $i = 1, 2$, the diagram of functors $\mathbb{H}(S_1) \to \mathbb{H}(T_1 \times T_2)$

$$ \begin{tikzcd} f_1^* A_1 \boxtimes f_2^* \mathbf{1}_{S_2} \arrow[r, "\mathbf{1}^*"'] & f_1^* A_1 \boxtimes \mathbf{1}_{T_2} \arrow[r, "u_r"'] & pr_1^{12,*} f_1^* A_1 \arrow[d, "m"'] \\ (f_1 \times f_2)^* (A_1 \boxtimes \mathbf{1}_{S_2}) \arrow[r, "u_r"'] & (f_1 \times f_2)^* pr_1^{12,*} A_1 \arrow[d, "\|"'] \end{tikzcd} $$

and the diagram of functors $\mathbb{H}(S_2) \times \mathbb{H}(T_1 \times T_2)$

$$ \begin{tikzcd} f_1^* \mathbf{1}_{S_1} \boxtimes f_2^* A_2 \arrow[r, "\mathbf{1}^*"'] & \mathbf{1}_{T_1} \boxtimes f_2^* A_2 \arrow[r, "u_r"'] & pr_2^{12,*} f_2^* A_2 \arrow[d, "m"'] \\ (f_1 \times f_2)^* (\mathbf{1}_{T_1} \boxtimes A_2) \arrow[r, "u_r"'] & (f_1 \times f_2)^* pr_2^{12,*} A_2 \arrow[d, "\|"'] \end{tikzcd} $$

are commutative.

We say that $(\boxtimes, m; u)$ is a *unitary external tensor structure* on $\mathbb{H}$.

(2) Let $(\boxtimes, m; u)$ and $(\boxtimes', m'; u')$ be unitary external tensor structures on $\mathbb{H}$. An *equivalence of unitary external tensor structures* $e : (\boxtimes, m; u) \xrightarrow{\sim} (\boxtimes', m'; u')$ is the datum of

* an equivalence of internal tensor structures $e : (\boxtimes, m) \xrightarrow{\sim} (\boxtimes', m')$,

* an isomorphism of unit sections $w : \mathbf{1} \xrightarrow{\sim} \mathbf{1}'$

satisfying the following additional condition:

(eq-uETS) For every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_1) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{tikzcd} A_1 \boxtimes' \mathbf{1}_{S_2} \arrow[r, "u_r'"'] & pr_1^{12,*} A_1 \arrow[d, "w"'] \\ A_1 \boxtimes \mathbf{1}_{S_2} \arrow[r, "e"'] & A_1 \boxtimes \mathbf{1}_{S_2} \arrow[d, "\uparrow"'] \end{tikzcd} $$

and the diagram of functors $\mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{tikzcd} \mathbf{1}_{S_1} \boxtimes' A_2 \arrow[r, "u_l'"'] & pr_2^{12,*} A_2 \arrow[d, "w"'] \\ \mathbf{1}_{S_1} \boxtimes A_2 \arrow[r, "e"'] & \mathbf{1}_{S_2} \boxtimes A_2 \arrow[d, "\uparrow"'] \end{tikzcd} $$

are commutative.

In the same spirit as for the previous two sections, we now describe how unit constraints can be translated between the internal and the external setting along the constructions of Lemma 2.10 and Lemma 2.11. This is the content of the following two results:

**Lemma 5.3.** Let $(\boxtimes, m)$ be an internal tensor structure on $\mathbb{H}$, and let $(\boxtimes, m^e)$ denote the external tensor structure defined from it as in Lemma 2.10. Suppose that we are given an internal unit constraint $u$ (with unit section $\mathbf{1}$) on $(\boxtimes, m)$. Then, associating

* to every $S_1, S_2 \in \mathcal{S}$, the natural isomorphism of functors $\mathbb{H}(S_1) \to \mathbb{H}(S_1 \times S_2)$

$$ u_r^e : A_1 \boxtimes \mathbf{1}_{S_2} := pr_1^{12,*} A_1 \otimes pr_2^{12,*} \mathbf{1}_{S_2} \xrightarrow{\mathbf{1}^*} pr_1^{12,*} A_1 \otimes \mathbf{1}_{S_1 \times S_2} \xrightarrow{u_r} pr_1^{12,*} A_1 $$

and the natural isomorphism of functors $\mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ u_l^e : \mathbf{1}_{S_2} \boxtimes A_2 := pr_1^{12,*} \mathbf{1}_{S_1} \otimes pr_2^{12,*} A_2 \xrightarrow{\mathbf{1}^*} \mathbf{1}_{S_1 \times S_2} \otimes pr_2^{12,*} A_2 \xrightarrow{u_l} pr_2^{12,*} A_2 $$
defines an external unit constraint $u^e$ with unit section $1$ on $(\boxtimes, m^e)$.

*Proof.* We have to show that the definition satisfies conditions (uETS-0) and (uETS-1).

We start from condition (uETS-0): given $S_1, S_2 \in \mathcal{S}$, we have to show that the diagram in $\mathbb{H}(S)$

<figure><img src="image_19.png" /></figure>

is commutative. Unwinding the various definitions, we obtain the more explicit diagram

<figure><img src="image_20.png" /></figure>

that we decompose as

<figure><img src="image_21.png" /></figure>

Here, the central piece is commutative by axiom (uITS-0) while the other pieces are commutative by naturality and by axiom (S-sect). Hence the outer part of the diagram is commutative as well.

We now treat condition (uEST-1): given two morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, $i = 1, 2$, we have to show that the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(T_1 \times T_2)$

<figure><img src="image_22.png" /></figure>

and the diagram of functors $\mathbb{H}(S_2) \times \mathbb{H}(T_1 \times T_2)$

<figure><img src="image_23.png" /></figure>
are commutative. We only check the commutativity of the first diagram; the case of the second diagram is analogous. Unwinding the various definitions, we obtain the more explicit diagram

<figure><img src="image_24.png" /></figure>

that we decompose as

<figure><img src="image_25.png" /></figure>

Here, the lower-right piece is commutative by axiom (uITS-1) while the other pieces are commutative by naturality and by axiom (S-sect). Hence the outer part of the diagram is commutative as well. □

**Lemma 5.4.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$, and let $(\otimes, m^i)$ denote the internal tensor structure defined from it as in Lemma 2.11. Suppose that we are given an external unit constraint $u$ (with unit section $\mathbf{1}$) on $(\boxtimes, m)$. Then, associating

• to every $S \in \mathcal{S}$, the two natural isomorphisms of functors $\mathbb{H}(S) \to \mathbb{H}(S)$

$$ u_r^i : A \otimes \mathbf{1}_S := \Delta_S^*(A \boxtimes \mathbf{1}_S) \xrightarrow{u_r} \Delta_S^* pr_1^{12,*} A = A, \quad u_l^i : \mathbf{1}_S \otimes A := \Delta_S^*(\mathbf{1}_S \boxtimes A) \xrightarrow{u_l} \Delta_S^* pr_2^{12,*} A = A $$

defines an external unit constraint $u^i$ (with unit section $\mathbf{1}$) on $(\boxtimes, m^i)$.

*Proof.* We have to show that the definition satisfies conditions (uITS-0) and (uITS-1).

We start from condition (uITS-0): given $S \in \mathcal{S}$, we have to show that the diagram in $\mathbb{H}(S)$

$$ \begin{tikzcd} \mathbf{1}_S \otimes \mathbf{1}_S \arrow[r, "\mathbf{1}_S"] & \phantom{\mathbf{1}_S} \\ (\mathbf{u}_r)_{\mathbf{S}} \arrow[u, "\mathbf{u}_l"] \arrow[r, "\mathbf{u}_l"] & \phantom{\mathbf{1}_S} \end{tikzcd} $$

is commutative. Unwinding the various definitions, we obtain the more explicit diagram

<figure><img src="image_26.png" /></figure>
that we decompose as

<figure><img src="image_27.png" /></figure>

Here, the upper piece is commutative by axiom (uITS-0) while the other pieces are commutative by axiom ($\mathcal{S}$-sect). Hence the outer part of the diagram is commutative as well.

We now treat condition (uITS-1): given a morphism $f: T \to S$ in $\mathcal{S}$, we have to show that the two diagrams of functors $\mathbb{H}(S) \to \mathbb{H}(T)$

<figure><img src="image_28.png" /></figure>

are commutative. We only check the commutativity of the first diagram; the case of the second diagram is analogous. Unwinding the various definitions, we obtain the more explicit diagram

<figure><img src="image_29.png" /></figure>

that we decompose as

<figure><img src="image_30.png" /></figure>

Here, the upper piece is commutative by axiom (uEST-1) while the other pieces are commutative by natur- ality and by Lemma 1.3. Hence the outer part of the diagram is commutative as well. □

**Proposition 5.5.** The constructions of Lemma 5.3 and Lemma 5.4 canonically define mutually inverse bijections between equivalence classes of unitary internal and external tensor structures with a given unit section **1** on $\mathbb{H}$.

*Proof.* We keep the same notation as in the proof of Proposition 2.12. Throughout the proof, we fix a section **1** of the $\mathcal{S}$-fibered category $\mathbb{H}$; the only isomorphism of unit section involved in the proof is the identity isomorphism $\text{id}_1: \mathbb{1} \xrightarrow{\sim} \mathbb{1}$.

Start from a unitary internal tensor structure $(\otimes, m; u)$, and let $u'$ denote the internal unit constraint on $(\otimes', m')$ obtained from $u$ by applying first Lemma 5.3 and then Lemma 5.4. We claim that the equivalence of
internal tensor structures $e : (\otimes, m) \xrightarrow{\sim} (\otimes', m')$ constructed in the first part of the proof of Proposition 2.12 satisfies condition (eq-uITS): given $S \in \mathcal{S}$, we have to show that the two diagrams of functors $\mathbb{H}(S) \to \mathbb{H}(S)$

<figure><img src="image_31.png" /></figure>

are commutative. We only check the commutativity of the diagram on the left; the case of the diagram on the right is completely analogous. Unwinding the various definitions, we obtain the more explicit diagram

<figure><img src="image_32.png" /></figure>

that we decompose as

<figure><img src="image_33.png" /></figure>

Here, the right-most piece is commutative by axiom (uITS-1) while the other pieces are commutative by naturality and by construction. This proves the claim.

Conversely, start from a unitary external tensor structure $(\boxtimes, m; u)$ on $\mathbb{H}$, and let $u'$ denote the external unit constraint on $(\boxtimes', m')$ obtained from $u$ by applying first Lemma 5.4 and then Lemma 5.3. We claim that the equivalence of external tensor structures $e : (\boxtimes', m') \xrightarrow{\sim} (\boxtimes, m)$ constructed in the second part of the proof of Proposition 2.12 satisfies condition (eq-uETS): given $S_1, S_2 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S_1) \to \mathbb{H}(S_1 \times S_2)$

<figure><img src="image_34.png" /></figure>

and the diagram of functors $\mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

<figure><img src="image_35.png" /></figure>
**Definition 6.3.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$, endowed with an external associativity constraint $a$ and an external commutativity constraint $c$.

We say that the constraints $a$ and $c$ are compatible if they satisfy the following condition:

<figure><img src="image_36.png" /></figure>

In this case, we say that $(\boxtimes, m; a, c)$ is a symmetric associative external tensor structure on $\mathbb{H}$.

**Lemma 6.4.** Let $(\otimes, m)$ be an internal tensor structure on $\mathbb{H}$, and let $(\boxtimes, m^e)$ denote the external tensor structure obtained from it via Lemma 2.10. Suppose that $(\otimes, m)$ is endowed with compatible internal associativity and commutativity constraints $a$ and $c$; let $a^e$ and $c^e$ denote the external associativity and commutativity constraints on $(\boxtimes, m)$ obtained from them via Lemma 3.3 and Lemma 3.4, respectively.

Then the external associativity and commutativity constraints $a^e$ and $c^e$ are compatible as well.

*Proof.* Let us check that the external constraints $a^e$ and $c^e$ satisfy condition (acETS): given $S_1, S_2, S_3 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(S_1 \times S_2 \times S_3)$

<figure><img src="image_37.png" /></figure>

is commutative. To this end, we decompose it as

<figure><img src="image_38.png" /></figure>

where, for notational convenience, we have set $B_i := \text{pr}_i^{123,*} A_i$, $i = 1, 2, 3$. The six unnamed natural isomorphisms are obtained by combining connection isomorphisms and internal monoidality isomorphisms along suitable paths; by Lemma 2.4, we know that the definitions do not depend on the chosen paths. Since the central rectangle in the latter diagram is commutative by axiom (acITS), it now suffices to show that the six lateral pieces are commutative as well. This follows by the usual trick; we omit the details. $\square$
**Lemma 6.5.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$, and let $(\otimes, m^i)$ denote the internal tensor structure obtained from it via Lemma 2.11. Suppose that $(\boxtimes, m)$ is endowed with compatible external associativity and commutativity constraints $a$ and $c$; let $a^i$ and $c^i$ denote the internal associativity and commutativity constraints on $(\otimes, m^i)$ obtained from them via Lemma 3.4 and Lemma 4.4, respectively.

Then the internal associativity and commutativity constraints $a^i$ and $c^i$ are compatible as well.

*Proof.* Let us check that the internal constraints $a^i$ and $c^i$ satisfy condition (acITS): for every $S \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{array}{ccc} (A \otimes B) \otimes C & \xrightarrow{c^i} & (B \otimes A) \otimes C \xrightarrow{a^i} B \otimes (A \otimes C) \\ \downarrow_{a^i} & & \downarrow_{c^i} \\ A \otimes (B \otimes C) & \xrightarrow{c^i} & (B \otimes C) \otimes A \xrightarrow{a^i} B \otimes (C \otimes A) \end{array} $$

is commutative. To this end, we decompose it as

<figure><img src="image_39.png" /></figure>

The six unnamed natural isomorphisms are obtained by combining connection isomorphisms and external monoidality isomorphisms along suitable paths; by Lemma 2.8, we know that the definitions do not depend on the chosen paths. Since the central rectangle in the latter diagram is commutative by axiom (acETS), it now suffices to show that the six lateral pieces are commutative as well. This follows by the usual trick; we omit the details. $\square$

## 6.2. Associativity versus unit.

**Definition 6.6.** Let $(\otimes, m)$ be an internal tensor structure on $\mathbb{H}$, endowed with an internal associativity constraint $a$ and an internal unit constraint $u$ (with unit section 1).

We say that the constraints $a$ and $u$ are compatible if they satisfy the following condition:

(auITS) For every $S \in \mathcal{S}$, the three diagrams of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{array}{ccc} (\mathbf{1}_S \otimes B) \otimes C & \xrightarrow{a} & \mathbf{1}_S \otimes (B \otimes C) \\ \downarrow_{u_l} & & \downarrow_{u_l} \\ B \otimes C & & \end{array} $$

$$ \begin{array}{ccc} (A \otimes \mathbf{1}_S) \otimes C & \xrightarrow{a} & A \otimes (\mathbf{1}_S \otimes C) \\ \downarrow_{u_r} & & \downarrow_{u_r} \\ A \otimes C & & \end{array} $$

$$ \begin{array}{ccc} (A \otimes B) \otimes \mathbf{1}_S & \xrightarrow{a} & A \otimes (B \otimes \mathbf{1}_S) \\ \downarrow_{u_r} & & \downarrow_{u_r} \\ A \otimes B & & \end{array} $$
are commutative.

In this case, we say that $(\otimes, m; a, u)$ is a *unitary associative internal tensor structure* on $\mathbb{H}$.

**Definition 6.7.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$, endowed with an external associativity constraint $a$ and an external unit constraint $u$ (with unit section **1**).

We say that the constraints $a$ and $u$ are compatible if they satisfy the following condition:

(auETS) For every $S_1, S_2, S_3 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(S_2 \times S_3)$

$$ \begin{array}{ccc} (\mathbf{1}_{S_1} \boxtimes A_2) \boxtimes A_3 & \xrightarrow{a} & \mathbf{1}_{S_1} \boxtimes (A_2 \boxtimes A_3) \\ \Big\downarrow_{u_l} & & \Big\downarrow_{u_l} \\ pr_2^{12,*} A_2 \boxtimes A_3 & \xrightarrow{m} & pr_{23}^{123,*} (A_2 \boxtimes A_3), \end{array} $$

the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_3) \to \mathbb{H}(S_1 \times S_3)$

$$ \begin{array}{ccc} (A_1 \boxtimes \mathbf{1}_{S_2}) \boxtimes A_3 & \xrightarrow{a} & A_1 \boxtimes (\mathbf{1}_{S_2} \boxtimes A_3) \\ \Big\downarrow_{u_r} & & \Big\downarrow_{u_l} \\ pr_1^{12,*} A_1 \boxtimes A_3 & \xrightarrow{m} & pr_{13}^{123,*} (A_1 \boxtimes A_3) \leftarrow_{m} A_1 \boxtimes pr_3^{23,*} A_3 \end{array} $$

and the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{array}{ccc} (A_1 \boxtimes A_2) \boxtimes \mathbf{1}_{S_3} & \xrightarrow{a} & A_1 \boxtimes (A_2 \boxtimes \mathbf{1}_{S_3}) \\ \Big\downarrow_{u_r} & & \Big\downarrow_{u_r} \\ pr_{23}^{123,*} (A_1 \boxtimes A_2) & \leftarrow_{m} & A_1 \boxtimes pr_2^{12,*} A_2 \end{array} $$

are commutative.

In this case, we say that $(\boxtimes, m; a, u)$ is a *unitary associative external tensor structure* on $\mathbb{H}$.

**Lemma 6.8.** Let $(\otimes, m)$ be an internal tensor structure on $\mathbb{H}$, and let $(\boxtimes, m)$ denote the external tensor structure obtained from it via Lemma 2.10. Suppose that $(\otimes, m)$ is endowed with compatible internal associativity and unit constraints $a$ and $u$, and let $a^e$ and $u^e$ denote the external associativity and unit constraints on $(\boxtimes, m)$ obtained from them via Lemma 3.3 and Lemma 5.3, respectively.

Then the external associativity and unit constraints $a^e$ and $u^e$ are compatible as well.

*Proof.* Let us check that the constraints $a^e$ and $u^e$ satisfy condition (auETS): for every $S_1, S_2, S_3 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(S_2 \times S_3)$

$$ \begin{array}{ccc} (\mathbf{1}_{S_1} \boxtimes A_2) \boxtimes A_3 & \xrightarrow{a^e} & \mathbf{1}_{S_1} \boxtimes (A_2 \boxtimes A_3) \\ \Big\downarrow_{u_l^e} & & \Big\downarrow_{u_l^e} \\ pr_2^{12,*} A_2 \boxtimes A_3 & \xrightarrow{m^e} & pr_{23}^{123,*} (A_2 \boxtimes A_3), \end{array} $$

the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_3) \to \mathbb{H}(S_1 \times S_3)$

$$ \begin{array}{ccc} (A_1 \boxtimes \mathbf{1}_{S_2}) \boxtimes A_3 & \xrightarrow{a^e} & A_1 \boxtimes (\mathbf{1}_{S_2} \boxtimes A_3) \\ \Big\downarrow_{u_r^e} & & \Big\downarrow_{u_l^e} \\ pr_1^{12,*} A_1 \boxtimes A_3 & \xrightarrow{m^e} & pr_{13}^{123,*} (A_1 \boxtimes A_3) \leftarrow_{m^e} A_1 \boxtimes pr_3^{23,*} A_3 \end{array} $$
and the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{array}{ccc} (A_1 \boxtimes A_2) \boxtimes \mathbf{1}_{S_3} & \xrightarrow{a^e} & A_1 \boxtimes (A_2 \boxtimes \mathbf{1}_{S_3}) \\ \downarrow u_r^e & & \downarrow u_r^e \\ pr_{23}^{123, *} (A_1 \boxtimes A_2) & \leftarrow_{m^e} & A_1 \boxtimes pr_2^{12, *} A_2 \end{array} $$

are commutative. We only show the commutativity of the first diagram; the cases of the second and third diagrams are completely analogous. Unwinding the various definitions, we obtain the more explicit diagram

$$ \begin{array}{ccc} (pr_1^{123,*} \mathbf{1}_{S_1} \otimes pr_2^{123,*} A_2) \otimes pr_3^{123,*} A_3 & \xrightarrow{a} & pr_1^{123,*} \mathbf{1}_{S_1} \otimes (pr_2^{123,*} A_2 \otimes pr_3^{123,*} A_3) \\ \multicolumn{3}{c}{\parallel} \\ (pr_{12}^{123,*} pr_1^{12,*} \mathbf{1}_{S_1} \otimes pr_{23}^{123,*} pr_2^{23,*} A_2) \otimes pr_3^{123,*} A_3 & \xrightarrow{m} & pr_1^{123,*} \mathbf{1}_{S_1} \otimes (pr_{23}^{123,*} pr_2^{23,*} A_2 \otimes pr_{23}^{123,*} pr_3^{23,*} A_3) \\ \multicolumn{3}{c}{\parallel} \\ pr_{12}^{123,*} (pr_1^{12,*} \mathbf{1}_{S_1} \otimes pr_2^{23,*} A_2) \otimes pr_3^{123,*} A_3 & \xrightarrow{1^*} & pr_1^{123,*} \mathbf{1}_{S_1} \otimes pr_{23}^{123,*} (pr_2^{23,*} A_2 \otimes pr_3^{23,*} A_3) \\ \multicolumn{3}{c}{\parallel} \\ pr_{12}^{123,*} (\mathbf{1}_{S_1 \times S_2} \otimes pr_2^{23,*} A_2) \otimes pr_3^{123,*} A_3 & \xrightarrow{u_l} & \mathbf{1}_{S_1 \times S_2 \times S_3} \otimes pr_{23}^{123,*} (pr_2^{23,*} A_2 \otimes pr_3^{23,*} A_3) \\ \multicolumn{3}{c}{\parallel} \\ pr_{12}^{123,*} pr_2^{23,*} A_2 \otimes pr_3^{123,*} A_3 & \xrightarrow{u_l} & pr_{23}^{123,*} (pr_2^{23,*} A_2 \otimes pr_3^{23,*} A_3) \\ \multicolumn{3}{c}{\parallel} \\ pr_2^{123,*} A_2 \otimes pr_3^{123,*} A_3 & \parallel & pr_{23}^{123,*} pr_2^{23,*} A_2 \otimes pr_{23}^{123,*} pr_3^{23,*} A_3 \end{array} $$

that we decompose as

<figure><img src="image_40.png" /></figure>

Here, the central triangle is commutative by axiom (auITS) while the remaining pieces are commutative by naturality and by axioms (S-sect) and (uITS-1). This proves the claim. $\square$
**Lemma 6.9.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$, and let $(\otimes, m)$ denote the internal tensor structure obtained from it via Lemma 2.11. Suppose that $(\boxtimes, m)$ is endowed with compatible external associativity and unit constraints $a$ and $u$, and let $a^i$ and $u^i$ denote the internal associativity and unit constraints on $(\otimes, m)$ obtained from them via Lemma 3.4 and Lemma 5.4, respectively.

Then the internal associativity and unit constraints $a^i$ and $u^i$ are compatible as well.

*Proof.* Let us check that the constraints $a^i$ and $u^i$ in the statement satisfy condition (auITS): for every $S \in \mathcal{S}$, we have to show that the three diagrams of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

are commutative. We only show the commutativity of the first diagram; the cases of the second and third diagram are completely analogous. Unwinding the various definitions, we obtain the more explicit diagram

<figure><img src="image_44.png" /></figure>

that we decompose as

<figure><img src="image_45.png" /></figure>

Here, the upper piece is commutative by axiom (auETS) while the remaining pieces are commutative by naturality, by Lemma 2.8 and by axiom (uETS-1). This proves the claim. $\square$

## 6.3. Commutativity versus unit.

**Definition 6.10.** Let $(\otimes, m)$ be an internal tensor structure on $\mathbb{H}$, endowed with an internal commutativity constraint $c$ and an internal unit constraint $u$ (with unit section 1).
We say that the constraints $c$ and $u$ are compatible if they satisfy the following condition: (cuITS) For every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S) \to \mathbb{H}(S)$

<figure><img src="image_46.png" /></figure>

is commutative.

In this case, we say that $(\otimes, m; c, u)$ is a *unitary symmetric internal tensor structure on $\mathbb{H}$*.

**Definition 6.11.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$, endowed with an external commutativity constraint $c$ and an external unit constraint $u$ (with unit section **1**).

We say that the constraints $c$ and $u$ are compatible if they satisfy the following condition: (cuETS) For every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

<figure><img src="image_47.png" /></figure>

is commutative.

In this case, we say that $(\boxtimes, m; c, u)$ is a *unitary symmetric external tensor structure on $\mathbb{H}$*.

**Lemma 6.12.** Let $(\boxtimes, m)$ be an internal tensor structure on $\mathbb{H}$, and let $(\boxtimes, m)$ denote the external tensor structure obtained from it via Lemma 2.10. Suppose that $(\boxtimes, m)$ is endowed with compatible internal commutativity and unit constraints $c$ and $u$, and let $c^e$ and $u^e$ denote the external commutativity and unit constraints on $(\boxtimes, m)$ obtained from them via Lemma 4.3 and Lemma 5.3, respectively.

Then the external commutativity and unit constraints $c^e$ and $u^e$ are compatible as well.

*Proof.* Let us check that the constraints $c^e$ and $u^e$ in the statement satisfy condition (cuETS): for every $S_1, S_2 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

<figure><img src="image_48.png" /></figure>

is commutative. Unwinding the various definitions, we obtain the more explicit diagram

<figure><img src="image_49.png" /></figure>
that we decompose as

<figure><img src="image_50.png" /></figure>

Here, the lower-right piece is commutative by axiom (cuITS) while the remaining pieces are commutative by naturality and by axiom (S-sect). This concludes the proof. □

**Lemma 6.13.** Let $(\boxtimes, m)$ be an external tensor structure on $\mathbb{H}$, and let $(\otimes, m)$ denote the internal tensor structure obtained from it via Lemma 2.11. Suppose that $(\boxtimes, m)$ is endowed with compatible external commutativity and unit constraints $c$ and $u$, and let $c^i$ and $u^i$ denote the internal commutativity and unit constraints on $(\otimes, m)$ obtained from them via Lemma 4.4 and Lemma 5.4, respectively.

Then the internal commutativity and unit constraints $c^i$ and $u^i$ are compatible as well.

*Proof.* Let us check that the constraints $c^i$ and $u^i$ in the statement satisfy condition (cuITS): for every $S \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S) \to \mathbb{H}(S)$

<figure><img src="image_51.png" /></figure>

is commutative. Unwinding the various definitions, we obtain the more explicit diagram

<figure><img src="image_52.png" /></figure>

that we decompose as

<figure><img src="image_53.png" /></figure>

Here, the upper-left piece is commutative by axiom (cuETS) while the remaining pieces are commutative by naturality. This concludes the proof. □
## 7. PROJECTION FORMULAE

In many concrete geometric situations, one is interested in monoidal fibered categories satisfying the so-called projection formulae. The example that we have in mind is that of stable homotopic 2-functors studied in [Ayo07a] and [Ayo07b]: in this case, the relevant definition is [Ayo07a, Defn. 2.3.1].

In this short section, we study the validity of projection formulae in the abstract setting of internal and external tensor structures. To this end, we need to introduce more structure on the base category $\mathcal{S}$.

**Hypothesis 7.1.** We assume that the category $\mathcal{S}$ is endowed with a distinguished subcategory $\mathcal{S}^{sm}$ with the following property: For every Cartesian square in $\mathcal{S}$

$$ \begin{array}{ccc} P' & \xrightarrow{g'} & P \\ | & \downarrow p' & \downarrow p \\ S' & \xrightarrow{g} & S \end{array} $$

with $p$ in $\mathcal{S}^{sm}$, the morphism $p'$ belongs to $\mathcal{S}^{sm}$ as well. We call smooth morphisms the morphisms in $\mathcal{S}^{sm}$.

**Example 7.2.** The motivating examples are when $\mathcal{S} = \mathrm{Var}_k$ is the category of (quasi-projective) algebraic varieties over a field $k$ and $\mathcal{S}^{sm}$ denotes its subcategory of smooth morphisms. The same discussion also applies when $\mathcal{S} = \mathrm{Sm}_k$ is the category of smooth (quasi-projective) $k$-varieties and $\mathcal{S}^{sm}$ denotes again its subcategory of smooth morphisms.

**Definition 7.3.** We say that the $\mathcal{S}$-fibered category $\mathbb{H}$ is geometric if it satisfies the following two conditions:

($\mathcal{S}^{sm}$-0) For every smooth morphism $p: P \to S$ in $\mathcal{S}$, the functor $p^*: \mathbb{H}(S) \to \mathbb{H}(P)$ admits a left-adjoint $p_*: \mathbb{H}(P) \to \mathbb{H}(S)$.

($\mathcal{S}^{sm}$-1) For every Cartesian square in $\mathcal{S}$

$$ \begin{array}{ccc} P' & \xrightarrow{g'} & P \\ | & \downarrow p' & \downarrow p \\ S' & \xrightarrow{g} & S \end{array} $$

where $p$ (and hence $p'$) is a smooth morphism, the natural transformation of functors $\mathbb{H}(P) \to \mathbb{H}(S')$

$$ p'_{\#} g'^* A \xrightarrow{\eta} p'_{\#} g'^* p^* p_{\#} A = p'_{\#} p'^* g^* p_{\#} A \xrightarrow{\epsilon} g^* p_{\#} A $$

is invertible.

Throughout this section, we fix a geometric $\mathcal{S}$-fibered category $\mathbb{H}$ with respect to $\mathcal{S}^{sm}$.

Following [Ayo07a, Defn. 2.3.1], we introduce the projection formulae in the setting of internal tensor structures:

**Definition 7.4.** We say that an internal tensor structure $(\otimes, m)$ on $\mathbb{H}$ is geometric if the following condition holds:

(ITS-pf) For every smooth morphism $p: P \to S$, the natural transformation of functors $\mathbb{H}(P) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ (7) \qquad p_{\#}(A \otimes p^* B) \xrightarrow{\eta} p_{\#}(p^* p_{\#} A \otimes p^* B) \xrightarrow{m} p_{\#} p^*(p_{\#} A \otimes B) \xrightarrow{\epsilon} p_{\#} A \otimes B $$

and the natural transformation of functors $\mathbb{H}(S) \times \mathbb{H}(P) \to \mathbb{H}(S)$

$$ p_{\#}(p^* A \otimes B) \xrightarrow{\eta} p_{\#}(p^* A \otimes p^* p_{\#} B) \xrightarrow{m} p_{\#} p^*(A \otimes p_{\#} B) \xrightarrow{\epsilon} A \otimes p_{\#} B $$

are invertible.
With some minor modifications, we obtain the analogous definition in the setting of external tensor structures:

**Definition 7.5.** We say that an external tensor structure $(\boxtimes, m)$ on $\mathbb{H}$ is geometric if the following condition holds:

(ETS-pf) For every choice of smooth morphisms $p_i : P_i \to S_i$, $i = 1, 2$, the natural transformation of functors $\mathbb{H}(P_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ (8) \qquad \begin{array}{ccc} (p_1 \times \mathrm{id}_{S_2})_{\#} (A_1 \boxtimes A_2) & & p_{1, \#} A_1 \boxtimes A_2 \\ \multicolumn{1}{c}{\searrow \eta} & \multicolumn{1}{c}{\uparrow \epsilon} & \\ (p_1 \times \mathrm{id}_{S_2})_{\#} (p_1^* p_{1, \#} A_1 \boxtimes A_2) & \xrightarrow{m} & (p_1 \times \mathrm{id}_{S_2})_{\#} (p_1 \times \mathrm{id}_{S_2})^* (p_{1, \#} A_1 \boxtimes A_2) \end{array} $$

and the natural transformation of functors $\mathbb{H}(S_1) \times \mathbb{H}(P_2) \to \mathbb{H}(S_1 \times S_2)$

$$ (\mathrm{id}_{S_1} \times p_2)_{\#} (A_1 \boxtimes A_2) \qquad A_1 \boxtimes p_{2, \#} A_2 \\ \multicolumn{1}{c}{\searrow \eta} \qquad \uparrow \epsilon \\ (\mathrm{id}_{S_1} \times p_2)_{\#} (A_1 \boxtimes p_2^* p_{2, \#} A_2) \xrightarrow{m} (\mathrm{id}_{S_1} \times p_2)_{\#} (\mathrm{id}_{S_1} \times p_2)^* (A_1 \boxtimes p_{2, \#} A_2) $$

are invertible.

**Remark 7.6.** Both in the setting of internal and external tensor structures, the property of being geometric is clearly stable under equivalence.

Here is the only result of the present section:

**Proposition 7.7.** Let $\mathbb{H}$ be a geometric $\mathcal{S}$-fibered category. Let $(\otimes, m^i)$ be an internal tensor structure on $\mathbb{H}$, and let $(\boxtimes, m^e)$ denote the external tensor structure corresponding to it under the bijection of Proposition 2.12. Then $(\otimes, m^i)$ is geometric (in the sense of Definition 7.4) if and only if $(\boxtimes, m^e)$ is geometric (in the sense of Definition 7.5).

*Proof.* In view of Remark 7.6 the statement of the proposition makes sense, and it suffices to check that the constructions of Lemma 2.10 and Lemma 2.11 between internal and external tensor structures preserve the property of being geometric. By symmetry, it suffices to show that the invertibility of (7) implies that of (8) and conversely.

Let $(\otimes, m)$ be a geometric internal tensor structure on $\mathbb{H}$, and let $(\boxtimes, m^e)$ denote the external tensor structure obtained from it via Lemma 2.10. By construction, the natural transformation (8) associated to $(\boxtimes, m^e)$ is the left vertical composite in the commutative diagram

$$ \begin{tikzcd}[column sep=2.8em, row sep=2.8em] % Adjust these values to change the layout
    (p_1 \times \mathrm{id}_{S_2})_{\#} (pr_1^{12*} A_1 \otimes pr_2^{12*} A_2) \arrow[r, "(p_1 \times \mathrm{id}_{S_2})_{\#} (pr_1^{12*} A_1 \otimes (p_1 \times \mathrm{id}_{S_2})^{*} pr_2^{12*} A_2")] & (p_1 \times \mathrm{id}_{S_2})_{\#} ((pr_1^{12*} A_1 \otimes (p_1 \times \mathrm{id}_{S_2})^{*} pr_2^{12*} A_2)) \arrow[r, "\eta"] & \\
    (p_1 \times \mathrm{id}_{S_2})_{\#} (pr_1^{12*} p_1^* p_{1, \#} A_1 \otimes pr_2^{12*} A_2) \arrow[r, "\parallel", "\sim"] & (p_1 \times \mathrm{id}_{S_2})_{\#} ((pr_1 \times \mathrm{id}_{S_2})^{*} pr_1^{12*} p_{1, \#} A_1 \otimes (p_1 \times \mathrm{id}_{S_2})^{*} pr_2^{12*} A_2) \arrow[r, "m"] & \\
    (p_1 \times \mathrm{id}_{S_2})_{\#} ((pr_1 \times \mathrm{id}_{S_2})^{*} pr_1^{12*} p_{1, \#} A_1 \otimes (p_1 \times \mathrm{id}_{S_2})^{*} pr_2^{12*} A_2) \arrow[r, "\sim", "\sim"] & (p_1 \times \mathrm{id}_{S_2})_{\#} (pr_1^{12*} A_1 \otimes (p_1 \times \mathrm{id}_{S_2})^{*} pr_2^{12*} A_2) \arrow[r, "\epsilon"] & \\
    (p_1 \times \mathrm{id}_{S_2})_{\#} (pr_1^{12*} p_{1, \#} A_1 \otimes pr_2^{12*} A_2) \arrow[r, "\epsilon"] & ((p_1 \times \mathrm{id}_{S_2})_{\#} pr_1^{12*} A_1 \otimes (p_1 \times \mathrm{id}_{S_2})^{*} pr_2^{12*} A_2) \arrow[r, "\sim", "\sim"] & \\
    (pr_1^{12*} p_{1, \#} A_1 \otimes pr_2^{12*} A_2) & &
\end{tikzcd} $$
where the three diagonal natural isomorphisms are induced by the natural isomorphism between functors
$\mathbb{H}(P_1) \to \mathbb{H}(S_1 \times S_2)$

$$
(p_1 \times \mathrm{id}_{S_2})_{\#} \mathrm{pr}_1^{12*} A_1 \xrightarrow{\sim} \mathrm{pr}_1^{12*} p_{1,\#} A_1
$$

obtained by applying axiom (S$^{sm}$-1) to the Cartesian square in S

$$
\begin{tikzcd}
& & P_1 \times S_2 \arrow[r, "pr_1^{12}"] \arrow[d, "p_1 \times \mathrm{id}_{S_2}"] & P_1 \arrow[d, "p_1"] \\
S_1 \times S_2 & & pr_1^{12} & S_1.
\end{tikzcd}
$$

Since the right vertical composite in the above diagram is invertible by definition (as (⊗, m) is assumed to be geometric), the left vertical composite must be invertible as well.

Conversely, let (⊗, m) be a geometric external tensor structure on H, and let (⊗, m^i) denote the internal tensor structure obtained from it via Lemma 2.11. By construction, the composite natural transformation (7) associated to (⊗, m^i) is the left side of the commutative diagram

<figure><img src="image_54.png" /></figure>

where the three diagonal natural isomorphisms are induced by the natural isomorphism between functors
$\mathbb{H}(P \times S) \to \mathbb{H}(S)$

$p_{\#}(\mathrm{id}_P, p)^* A \xrightarrow{\sim} \Delta_S^*(p \times \mathrm{id}_S)_{\#} A$

obtained by applying axiom (S$^{sm}$-1) to the Cartesian square

$$
\begin{tikzcd}
& & P \arrow[r, "(\mathrm{id}_P, p)"'] & P \times S \arrow[d, "p"] \\
S & & \Delta_S & S \times S.
\end{tikzcd}
$$

Since the right vertical composite in the above diagram is invertible by definition (as (⊗, m) is assumed to be geometric), the composite on the left side must be invertible as well. □

8. TENSOR STRUCTURES ON MORPHISMS OF FIBERED CATEGORIES

Throughout this section, we fix two $\mathcal{S}$-fibered categories $\mathbb{H}_1$ and $\mathbb{H}_2$ as well as a morphism of $\mathcal{S}$-fibered categories $R : \mathbb{H}_1 \to \mathbb{H}_2$; following the notation of Section 1, for every morphism $f : T \to S$ in $\mathcal{S}$ we let $\theta = \theta_f : f^* \circ R_S \xrightarrow{\sim} R_T \circ f^*$ denote the corresponding $R$-transition isomorphism.

We are interested in studying how the notions introduced in the previous sections behave under the
morphism $R$. To this end, we introduce analogues of tensor structures and their constraints for morphisms,
both in the internal and in the external setting, and we compare them. The final outcome will be our second main result:

**Theorem 8.1.** For $j = 1, 2$, let $(\otimes_j, m_j^i)$ be a (unitary, symmetric, associative) internal tensor structure on $\mathbb{H}_j$, and let $(\boxtimes_j, m_j^e)$ denote the (unitary, symmetric, associative) external tensor structure corresponding to it under the bijection of Theorem 6.1.

Then there exist canonical bijections between (unitary, symmetric, associative) internal tensor structures on $R$ (with respect to $(\otimes_1, m_1^i)$ and $(\otimes_2, m_2^i)$) and (unitary, symmetric, associative) external tensor structures on $R$ (with respect to $(\boxtimes_1, m_1^e)$ and $(\boxtimes_2, m_2^e)$).

By analogy with the case of tensor structures on single fibered categories, the first step of the proof consists in constructing explicit mutually inverse bijections between internal and external tensor structures on $R$: this is achieved in the first subsection. The second step is to show that such a bijection respects each possible additional constraint: this is achieved in the next three subsections. As the reader will easily recognize, both the definitions and the arguments collected below are formally analogous to those of the previous sections.

**8.1. Internal and external tensor structures on morphisms.** We start by introducing the notion of internal tensor structure on the morphism $R$. This is the natural extension of the usual notion of monoidal morphism between monoidal categories to the fibered setting.

**Definition 8.2.** Let $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$ be internal tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$, respectively. An internal tensor structure $\rho$ on $R$ (with respect to $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$) is the datum of

* for every $S \in \mathcal{S}$, a natural isomorphism of functors $\mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(S)$

$$ \rho = \rho_S : R_S(A) \otimes_2 R_S(B) \xrightarrow{\sim} R_S(A \otimes_1 B), $$

called the *internal R-monoidality isomorphism at S*

satisfying the following condition:

(mor-ITS) For every morphism $f: T \to S$ in $\mathcal{S}$, the natural diagram of functors $\mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(T)$

$$ \begin{tikzcd} f^* R_S(A) \otimes_2 f^* R_S(B) \arrow[r, "m_2"] \arrow[d, "\theta"] & f^*(R_S(A) \otimes_2 R_S(B)) \arrow[r, "\rho"] \arrow[d, "\theta"] \\ R_T(f^*A) \otimes_2 R_T(f^*B) \arrow[r, "\rho"] & R_T(f^*A \otimes_1 f^*B) \arrow[r, "m_1"] \arrow[d, "\theta"'] \end{tikzcd} $$

is commutative.

The meaning of axiom (mor-ITS) is explained by the following result:

**Lemma 8.3.** For $j = 1, 2$ let $(\otimes_j, m_j)$ be an internal tensor structure on $\mathbb{H}_j$; let $f: T \to S$ be a morphism in $\mathcal{S}$. For every composable triple of composable tuples in $\mathcal{S}$

$$ \underline{g} = (g_1, \dots, g_r), \quad \underline{h} = (h_1, \dots, h_s), \quad \underline{l} = (l_1, \dots, l_t) $$

satisfying $l_t \circ \dots \circ l_1 \circ h_s \circ \dots \circ h_1 \circ g_r \circ \dots \circ g_1 = f$, consider the two functors

$$ F_{(g,h,l)}^{\otimes_1 R}, F_{(g,h,l)}^{\otimes_2 R} : \mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(T) $$

defined by the formulas

$$ F_{(g,h,l)}^{\otimes_1 R}(A, B) := \underline{g}^* R_{T'}(\underline{h}^*(\underline{l}^* A \otimes_1 \underline{l}^* B)) $$

and

$$ F_{(g,h,l)}^{\otimes_2 R}(A, B) := \underline{g}^*(\underline{h}^* R_{T''}(\underline{l}^* A) \otimes_2 \underline{h}^* R_{T''}(\underline{l}^* B)), $$

where $T'$ and $T''$ denote the domains of $h_1$ and $l_1$, respectively.
Then, for any two given composable triples of composable tuples $(g^{(1)}, \underline{h}^{(1)}, \underline{l}^{(1)})$ and $(g^{(2)}, \underline{h}^{(2)}, \underline{l}^{(2)})$ as above, and for each choice of $\epsilon_1, \epsilon_2 \in \{1, 2\}$, all possible isomorphisms of functors $\mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(T)$

$$F_{(g^{(1)}, \underline{h}^{(1)}, \underline{l}^{(1)})}^{\otimes \epsilon_1, R} \xrightarrow{\sim} F_{(g^{(2)}, \underline{h}^{(2)}, \underline{l}^{(2)})}^{\otimes \epsilon_2, R}$$

obtained by composing connection isomorphisms, internal monoidality isomorphisms, connection R-isomorphisms and internal R-monoidality isomorphisms (and inverses thereof) coincide.

Proving this result directly would be extremely intricate, due to the presence of four different types of natural isomorphisms. To circumvent the difficulty, we use a variant of Construction 2.5 based on Construction 1.10:

**Construction 8.4.** Keep the notation of Lemma 2.4. Consider the two $(\mathcal{S} \times \mathcal{I})$-fibered categories $\mathbb{H}_1^{\otimes 1}$ and $\mathbb{H}_2^{\otimes 2}$ obtained via Construction 2.5. Then, following Notation 1.9, we can define a morphism of $(\mathcal{S} \times \mathcal{I})$-fibered categories $R^\rho : \mathbb{H}_1^{\otimes 1} \to \mathbb{H}_2^{\otimes 2}$ as follows:

* For every $S \in \mathcal{S}$, we set $R_{(S;1)}^\rho(A, B) := (R_S(A), R_S(B))$ and $R_{(S;2)}^\rho(A) := R_S(A)$.

* Given a morphism $\phi$ in $\mathcal{S} \times \mathcal{I}$, we define the $R^\rho$-transition isomorphism $\theta_\phi$ according to the type of $\phi$:

- if $\phi = (f; 1) : (T; 1) \to (S; 1)$, we set

$$\theta_{\phi}(A) : (f^* R_S(A), f^* R_S(B)) \xrightarrow{(\theta_f, \theta_f)} (R_T(f^* A), R_T(f^* B));$$

- if $\phi = (f; 2) : (T; 2) \to (S; 2)$, we set

$$\theta_{\phi}(A) : f^* R_S(A) \xrightarrow{\theta_f} R_T(f^* A);$$

- if $\phi = r_S \circ (f; 2)$, we set

$$\theta_{\phi}(A, B) : f^*(R_S(A) \otimes_2 R_S(B)) \xrightarrow{\rho_S} f^* R_S(A \otimes_1 B).$$

Let us check that this construction indeed defines a morphism of $(\mathcal{S} \times \mathcal{I})$-fibered categories: given two composable morphisms $\phi : T \to S$ and $\psi : S \to V$ in $\mathcal{S} \times \mathcal{I}$, we have to show that the diagram

$$\begin{tikzcd}
(\psi\phi)^* R_V^\rho(A) \arrow[r, "\theta_{\psi\phi}^\rho"] & R_T^\rho((\psi\phi)^*A) \arrow[d, "\cong"] \\
\phi^*\psi^* R_V^\rho(A) \arrow[r, "\theta_\phi^\rho"] & \phi^* R_S^\rho(\psi^*A) \arrow[r, "\theta_\psi^\rho"] & R_T^\rho(\phi^*\psi^*A)
\end{tikzcd}$$

is commutative. We have the following possibilities, indexed by pairs of composable morphisms $f : T \to S$ and $g : S \to V$ in $\mathcal{S}$:

(1) If $\phi = (f; 1)$ and $\psi = (g; 1)$, we obtain the diagram

$$\begin{tikzcd}
((gf)^*R_V(A), (gf)^*R_V(B)) \arrow[r, "(\theta, \theta)"'] & (R_T((gf)^*A), R_T((gf)^*B)) \arrow[d, "\cong"] \\
(f^*g^*R_V(A), f^*g^*R_V(B)) \arrow[r, "(\theta, \theta)"'] & (f^*R_S(g^*A), f^*R_S(g^*B)) \arrow[r, "(\theta, \theta)"'] & (R_T(f^*g^*A), R_T(f^*g^*B))
\end{tikzcd}$$

which is commutative by axiom (mor-$\mathcal{S}$-fib).
As the reader may expect, we have the following analogue of Lemma 2.4:

**Lemma 8.6.** For $j = 1, 2$ let $(\boxtimes_j, m_j)$ be an external tensor structure on $\mathbb{H}_j$ and, for $i = 1, 2$, let $f_i : T_i \to S_i$ be a morphism in $\mathcal{S}$. Given, for $i = 1, 2$, a composable triple of composable tuples in $\mathcal{S}$

$$ g_i = (g_{i,1}, \dots, g_{i,r}), \quad h_i = (h_{i,1}, \dots, h_{i,s}), \quad l_i = (l_{i,1}, \dots, l_{i,t}) $$

satisfying $l_{i,t} \circ \dots \circ l_{i,1} \circ h_{i,s} \circ \dots \circ h_{i,1} \circ g_{i,r} \circ \dots \circ g_{i,1} = f_i$, consider the two functors

$$ F_{(g_1, h_1, l_1), (g_2, h_2, l_2)}^{\boxtimes_1, R}, \quad F_{(g_1, h_1, l_1), (g_2, h_2, l_2)}^{\boxtimes_2, R} : \mathbb{H}_1(S_1) \times \mathbb{H}_1(S_2) \to \mathbb{H}_2(T_1 \times T_2) $$

defined by the formulas

$$ F_{(g_1, h_1, l_1), (g_2, h_2, l_2)}^{\boxtimes_1, R}(A_1, A_2) := (g_1 \times g_2)^* R_{T'_1 \times T'_2}((h_1 \times h_2)^* (l_1^* A_1 \boxtimes_1 l_2^* A_2)) $$

and

$$ F_{(g_1, h_1, l_1), (g_2, h_2, l_2)}^{\boxtimes_2, R}(A_1, A_2) := (g_1 \times g_2)^* (h_1^* R_{T'_1} (l_1^* A_1) \boxtimes_2 h_2^* R_{T'_2} (l_2^* A_2)) $$

where, for $i = 1, 2$, $T'_i$ and $T''_i$ denote the domain of $h_{i,1}$ and $l_{i,1}$, respectively.

Then, for any two given composable triples of composable tuples $(g_i^{(1)}, h_i^{(1)}, l_i^{(1)})$ and $(g_i^{(2)}, h_i^{(2)}, l_i^{(2)})$ as above, $i = 1, 2$, and for any choice of numbers $\epsilon_1, \epsilon_2 \in \{1, 2\}$, all isomorphisms of functors $\mathbb{H}_1(S_1) \times \mathbb{H}_1(S_2) \to \mathbb{H}_2(T_1 \times T_2)$

$$ F_{(g_1^{(1)}, h_1^{(1)}, l_1^{(1)}), (g_2^{(1)}, h_2^{(1)}, l_2^{(1)})}^{\boxtimes_{\epsilon_1}, R}(A_1, A_2) \xrightarrow{\sim} F_{(g_1^{(2)}, h_1^{(2)}, l_1^{(2)}), (g_2^{(2)}, h_2^{(2)}, l_2^{(2)})}^{\boxtimes_{\epsilon_2}, R}(A_1, A_2) $$

obtained by composing connection isomorphisms, external monoidality isomorphisms, R-transition isomorphisms and external R-monoidality isomorphisms (and inverses thereof) coincide.

**Construction 8.7.** Keep the notation of Lemma 8.6. Consider the two $(\mathcal{S}^2 \times \mathcal{I})$-fibered categories $\mathbb{H}_1^{\boxtimes_1}$ and $\mathbb{H}_2^{\boxtimes_2}$ obtained via Construction 2.9. Then, following Notation 1.9, we can define a morphism of $(\mathcal{S}^2 \times \mathcal{I})$-fibered categories $R^\rho : \mathbb{H}_1^{\boxtimes_1} \to \mathbb{H}_2^{\boxtimes_2}$ as follows:

* For every $S_1, S_2 \in \mathcal{S}$, we set $R^\rho_{(S_1, S_2; 1)}(A_1, A_2) := (R_{S_1}(A_1), R_{S_2}(A_2))$ and $R^\rho_{(S_1, S_2; 2)}(A) := R_{S_1 \times S_2}(A)$.

* Given a morphism $\phi$ in $\mathcal{S}^2 \times \mathcal{I}$, we define the $R^\rho$-transition isomorphism $\theta_\phi$ according to the type of $\phi$:

$$ - \text{ if } \phi = (f_1, f_2; 1) : (T_1, T_2; 1) \to (S_1, S_2; 1), \text{ we set} $$

$$ \theta_\phi(A) : (f_1^* R_{S_1}(A_1), f_2^* R_{S_2}(A_2)) \xrightarrow{(\theta_{f_1}, \theta_{f_2})} (R_{T_1}(f_1^* A_1), R_{T_2}(f_2^* A_2)); $$

$$ - \text{ if } \phi = (f_1, f_2; 2) : (T_1, T_2; 2) \to (S_1, S_2; 2), \text{ we set} $$

$$ \theta_\phi(A) : (f_1 \times f_2)^* R_{S_1 \times S_2}(A_1) \xrightarrow{\theta_{f_1 \times f_2}} R_{T_1 \times T_2}((f_1 \times f_2)^* A); $$

$$ - \text{ if } \phi = r_{S_1, S_2} \circ (f_1, f_2; 2), \text{ we set} $$

$$ \theta_\phi(A_1, A_2) : (f_1 \times f_2)^* (R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2)) \xrightarrow{\rho_{S_1, S_2}} (f_1 \times f_2)^* R_{S_1 \times S_2}(A_1 \boxtimes_1 A_2). $$

Arguing in the same way as in Construction 8.4, one can check that this construction indeed defines a morphism of $(\mathcal{S}^2 \times \mathcal{I})$-fibered categories; we omit the details.

*Proof of Lemma 8.6.* As a consequence of Construction 8.7, this is just a particular case of Lemma 1.8. $\square$

**Remark 8.8.** In the following, we consider $\mathcal{S}$-fibered categories $\mathbb{H}_1, \mathbb{H}_2$ and $\mathbb{H}_3$ as well as morphisms of $\mathcal{S}$-fibered categories $R_1 : \mathbb{H}_1 \to \mathbb{H}_2$ and $R_2 : \mathbb{H}_2 \to \mathbb{H}_3$.
(1) Suppose that $\mathbb{H}_1 = \mathbb{H}_2 = \mathbb{H}$ and $R_1 = \text{id}_{\mathbb{H}}$. By definition, an internal tensor structure on $\text{id}_{\mathbb{H}}$ with respect to two given internal tensor structures $(\otimes', m')$ and $(\otimes, m)$ is the same as an equivalence of internal tensor structures $e : (\otimes, m) \xrightarrow{\sim} (\otimes', m')$.

(2) For $j = 1, 2, 3$ let $(\otimes_j, m_j)$ be an internal tensor structure on $\mathbb{H}_j$. Given internal tensor structures $\rho_1$ on $R_1$ (with respect to $\otimes_1$ and $\otimes_2$) and $\rho_2$ on $R_2$ (with respect to $\otimes_2$ and $\otimes_3$), we obtain an obvious composite internal tensor structure $\rho_2 \circ \rho_1$ on the composite morphism $R_2 \circ R_1$ (with respect to $\otimes_1$ and $\otimes_3$).

(3) For $j = 1, 2$ let $e_j : (\otimes_j, m_j) \xrightarrow{\sim} (\otimes'_j, m'_j)$ be an equivalence of internal tensor structures on $\mathbb{H}_j$, for simplicity, set $R := R_1$. Combining the previous two observations, we see that the pair $(e_1, e_2)$ determines a canonical bijection between internal tensor structures on $R$ with respect to $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$ and internal tensor structures on $R$ with respect to $(\otimes'_1, m'_1)$ and $(\otimes'_2, m'_2)$. In practice, this means that every question concerning the existence of internal tensor structures on the morphism $R$ only depends on the equivalence classes of the chosen internal tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$ in a very precise way.

Analogous considerations hold in the setting of external tensor structures.

The following two results explain how to translate an internal tensor structure on $R$ into an external tensor structure on it and conversely:

**Lemma 8.9.** For $j = 1, 2$, let $(\otimes_j, m_j)$ be an internal tensor structure on $\mathbb{H}_j$, and let $(\boxtimes_j, m_j^e)$ denote the external tensor structure obtained from it via Lemma 2.10. Suppose that we are given an internal tensor structure $\rho$ on $R$ (with respect to $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$). Then, associating

• to every $S_1, S_2 \in \mathcal{S}$, the natural isomorphism of functors $\mathbb{H}_1(S_1) \times \mathbb{H}_1(S_2) \to \mathbb{H}_2(S_1 \times S_2)$

$$ \rho^e = \rho_{S_1, S_2}^e : R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2) \xrightarrow{\sim} R_{S_1 \times S_2}(A_1 \boxtimes_1 A_2) $$

defined by taking the composite

$$ \begin{tikzcd} R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2) \arrow[r] \arrow[d, "\cong"] & R_{S_1 \times S_2}(A_1 \boxtimes_1 A_2) \arrow[d, "\cong"'] \\ pr_1^{12,*} R_{S_1}(A_1) \boxtimes_2 pr_2^{12,*} R_{S_2}(A_2) \arrow[r, "\theta"] & R_{S_1 \times S_2}(pr_1^{12,*} A_1) \boxtimes_2 R_{S_1 \times S_2}(pr_2^{12,*} A_2) \arrow[r, "\rho"] & R_{S_1 \times S_2}(pr_1^{12,*} A_1 \boxtimes_1 pr_2^{12,*} A_2) \end{tikzcd} $$

defines an external tensor structure $\rho^e$ on $R$ (with respect to $(\boxtimes_1, m_1^e)$ and $(\boxtimes_2, m_2^e)$).

*Proof.* We need to check that the definition satisfies condition (mor-ETS): given morphisms $f_i : T_1 \to S_i$ in $\mathcal{S}$, $i = 1, 2$, we have to show that the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(T_1 \times T_2)$

$$ \begin{tikzcd} f_1^* R_{S_1}(A_1) \boxtimes_2 f_2^* R_{S_2}(A_2) \arrow[r, "m_2^e"] \arrow[d, "\cong", "\theta"] & (f_1 \times f_2)^*(R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2)) \arrow[r, "\rho^e"] \arrow[d, "\cong", "\theta"] & (f_1 \times f_2)^* R_{S_1 \times S_2}(A_1 \boxtimes_1 A_2) \\ R_{T_1}(f_1^* A_1) \boxtimes_2 R_{T_2}(f_2^* A_2) \arrow[r, "\rho^e"] & R_{T_1 \times T_2}(f_1^* A_1 \boxtimes_1 f_2^* A_2) \arrow[r, "m_1^e"] & R_{T_1 \times T_2}((f_1 \times f_2)^*(A_1 \boxtimes_1 A_2)) \end{tikzcd} $$
is commutative. To this end, we decompose it as

<figure><img src="image_55.png" /></figure>

where, for notational convenience, we write $B_r := pr_r^{12,*} A_r$, $i = 1, 2$. The six unnamed isomorphisms are obtained by combining connection isomorphisms, internal monoidality isomorphisms and $R$-isomorphisms along suitable paths; by Lemma 8.3, we know that the definitions are independent of the chosen paths. Since the central rectangle in the latter diagram is commutative by axiom (mor-ITS), it now suffices to show that the six lateral pieces are commutative as well. This follows by the usual trick; we omit the details. $\square$

**Lemma 8.10.** For $j = 1, 2$, let $(\boxtimes_j, m_j)$ be an internal tensor structure on $\mathbb{H}_j$, and let $(\otimes_j, m_j^i)$ denote the external tensor structure obtained from it via Lemma 2.11. Suppose that we are given an external tensor structure $\rho$ on $R$ (with respect to $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$). Then, associating

* to every $S \in \mathcal{S}$, the natural isomorphism of functors $\mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(S)$

$$ (9) \qquad \rho^i = \rho_S^i : R_S(A) \otimes_2 R_S(B) \xrightarrow{\sim} R_S(A \otimes_1 B) $$

defined by taking the composite

$$ \begin{tikzcd} R_S(A) \otimes_2 R_S(B) \arrow[r] \arrow[d, "\Delta_S^*(R_S(A) \boxtimes_2 R_S(B))"] & R_S(A \otimes_1 B) \arrow[d, "\Delta_S^*(A \boxtimes_1 B)"] \\ \Delta_S^*(A \boxtimes_2 R_S(B)) & \Delta_S^*(A \boxtimes_1 B) \arrow[r, "\rho"] \arrow[d, "\theta"] & R_S(\Delta_S^*(A \boxtimes_1 B)) \arrow[d, "\theta"'] \end{tikzcd} $$

defines an internal tensor structure $\rho^e$ on $R$ (with respect to $(\otimes_1, m_1^i)$ and $(\otimes_2, m_2^i)$).

*Proof.* We need to check that the definition satisfies condition (mor-ITS): given a morphism $f: T \to S$ in $\mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ \begin{tikzcd} f^*R_S(A) \otimes_2 f^*R_S(B) \arrow[r, "m_2^i"] \arrow[d, "\theta"] & f^*(R_S(A) \otimes_2 R_S(B)) \arrow[r, "\rho^i"] \arrow[d, "\theta"] & f^*R_S(A \otimes_1 B) \arrow[d, "m_1^i"] \\ R_T(f^*A) \otimes_2 R_T(f^*B) \arrow[r, "\rho^i"] \arrow[d, "\theta"] & R_T(f^*A \otimes_1 f^*B) \arrow[r, "m_1^i"] \arrow[d, "\theta"] & R_T(f^*(A \otimes_1 B)) \arrow[d, "m_2^i"] \end{tikzcd} $$
is commutative. To this end, we decompose it as

<figure><img src="image_56.png" /></figure>

Here, the six unnamed natural isomorphisms are obtained by taking composites of connection isomorphisms, external monoidality isomorphisms and $R$-isomorphisms along suitable paths; by Lemma 8.6, we know that the definitions are independent of the chosen paths. Since the central rectangle is commutative by axiom (mor-ETS), it now suffices to show that the six lateral pieces are commutative as well. This follows by the usual trick; we omit the details. $\square$

**Proposition 8.11.** For $j = 1, 2$, let $(\otimes_j, m_j^i)$ be an internal tensor structure on $\mathbb{H}_i$, and let $(\boxtimes_j, m_j^e)$ denote the external tensor structure corresponding to it (modulo equivalence) via the bijection of Proposition 2.12. Then the constructions of Lemma 8.9 and Lemma 8.10 define mutually inverse bijections between internal tensor structures on $R$ (with respect to $(\otimes_1, m_1^i)$ and $(\otimes_2, m_2^i)$) and external tensor structures on $R$ (with respect to $(\boxtimes_1, m_1^e)$ and $(\boxtimes_2, m_2^e)$).

*Proof.* Start from two internal tensor structures $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$ and an internal tensor structure $\rho$ on $R$ (with respect to $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$). For $j = 1, 2$, let $(\otimes'_j, m'_j)$ denote the internal tensor structure on $\mathbb{H}_j$ obtained from $(\otimes_j, m_j)$ by applying first Lemma 2.10 and then Lemma 2.11; moreover, let $e_j : (\otimes_j, m_j) \xrightarrow{\sim} (\otimes'_j, m'_j)$ denote the equivalence of internal tensor structures constructed in Proposition 2.12. Finally, let $\rho'$ denote the internal tensor structure on $R$ (with respect to $(\otimes'_1, m'_1)$ and $(\otimes'_2, m'_2)$) obtained from $\rho$ by applying first Lemma 8.9 and then Lemma 8.10. We have to show that, for every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(S)$

$$ \begin{array}{ccc} R_S(A) \otimes_2 R_S(B) & \xrightarrow{\rho} & R_S(A \otimes_1 B) \\ \big|_{e_2} & & \big|_{e_1} \\ R_S(A) \otimes'_2 R_S(B) & \xrightarrow{\rho'} & R_S(A \otimes'_1 B) \end{array} $$

is commutative. Unwinding the various constructions, we obtain the more explicit diagram

$$ \begin{array}{ccc} R_S(A) \otimes_2 R_S(B) & \xrightarrow{\rho} & R_S(A \otimes_1 B) \\ \multicolumn{3}{c}{\parallel} \\ \Delta_S^* pr_1^{12,*} R_S(A) \otimes_2 \Delta_S^* pr_2^{12,*} R_S(B) & R_S(\Delta_S^* pr_1^{12,*} A \otimes_1 \Delta_S^* pr_2^{12,*} B) & \\ \multicolumn{3}{c}{\parallel} \\ \Delta_S^*(pr_1^{12,*} R_S(A) \otimes_2 pr_2^{12,*} R_S(B)) & R_S(\Delta_S^*(pr_1^{12,*} A \otimes_1 pr_2^{12,*} B)) & \\ \multicolumn{3}{c}{\theta} \\ \Delta_S^*(R_{S \times S}(pr_1^{12,*} A) \otimes_2 R_{S \times S}(pr_2^{12,*} B)) & \xrightarrow{\rho} & \Delta_S^* R_{S \times S}(pr_1^{12,*} A \otimes_1 pr_2^{12,*} B) \end{array} $$
that we decompose as

<figure><img src="image_57.png" /></figure>

Here, the lower-right piece is commutative by axiom (mor-ITS) while the other pieces are commutative by naturality and by axiom (mor-$\mathcal{S}$-fib). This proves the claim.

Conversely, start from two external tensor structures $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$ and an external tensor structure $\rho$ on $R$ (with respect to $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$). For $j = 1, 2$, let $(\boxtimes'_j, m'_j)$ denote the external tensor structure obtained from $(\boxtimes_j, m_j)$ by applying first Lemma 2.11 and then Lemma 2.10; moreover, let $e_j : (\boxtimes'_j, m'_j) \xrightarrow{\sim} (\boxtimes_j, m_j)$ denote the equivalence of external tensor structures constructed in Proposition 2.12. Finally, let $\rho'$ denote the external tensor structure on $R$ (with respect to $(\boxtimes'_1, m'_1)$ and $(\boxtimes'_2, m'_2)$) obtained from $\rho$ by applying first Lemma 8.10 and then Lemma 8.9. We have to show that, for every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}_1(S_1) \times \mathbb{H}_1(S_2) \to \mathbb{H}_2(S_1 \times S_2)$

$$
\begin{array}{ccc}
R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2) & \xrightarrow{\rho} & R_{S_1 \times S_2}(A_1 \boxtimes_1 A_2) \\
\multicolumn{2}{c}{e_2 \uparrow} & \multicolumn{2}{c}{e_1 \uparrow} \\
R_{S_1}(A_1) \boxtimes'_2 R_{S_2}(A_2) & \xrightarrow{\rho'} & R_{S_1 \times S_2}(A_1 \boxtimes'_1 A_2)
\end{array}
$$

is commutative. Unwinding the various constructions, we obtain the more explicit diagram

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2) \arrow[r, "\rho"] & R_{S_1 \times S_2}(A_1 \boxtimes_1 A_2) \arrow[||-]
  \arrow[||-]
  \arrow[r, "\rho'"] & R_{S_1 \times S_2}(A_1 \boxtimes'_1 A_2) \\
  \Delta_{S_1 \times S_2}^*(pr_1^{12} \times pr_2^{12})^*(R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2)) & R_{S_1 \times S_2}(\Delta_{S_1 \times S_2}^*(pr_1^{12} \times pr_2^{12})^*(A_1 \boxtimes_1 A_2)) \\
  \Delta_{S_1 \times S_2}^*(pr_1^{12,*} R_{S_1}(A_1) \boxtimes_2 pr_2^{12,*} R_{S_2}(A_2)) & R_{S_1 \times S_2}(\Delta_{S_1 \times S_2}^*(pr_1^{12,*} A_1 \boxtimes_1 pr_2^{12,*} A_2)) \\
  \Delta_{S_1 \times S_2}^*(R_{S_1 \times S_2}(pr_1^{12,*} A_1) \boxtimes_2 R_{S_1 \times S_2}(pr_2^{12,*} A_2)) & \rho \arrow[r, "\rho"] & \Delta_{S_1 \times S_2}^* R_{(S_1 \times S_2)^2}(pr_1^{12,*} A_1 \boxtimes_1 pr_2^{12,*} A_2)
\end{tikzcd}
$$
that we decompose as

<figure><img src="image_58.png" /></figure>

Here, the lower-left piece is commutative by axiom (mor-ETS) while the other pieces are commutative by naturality and by axiom (mor-$\mathcal{S}$-fib). This proves the claim and concludes the proof. □

**8.2. Associative, symmetric, unitary morphisms.** We now introduce the notions of associativity, commutativity and unitarity for internal and external tensor structures on morphisms, and we compare the two versions of each of them. This task is relatively easy since it does not involve constructing new structures but only amounts to checking some natural properties. This is done in Lemma 8.14 and Lemma 8.15 (associativity), Lemma 8.18 and Lemma 8.19 (symmetry), Lemma 8.22 and Lemma 8.23 (unitarity). Then Theorem 8.1 follows combining Proposition 8.11 with these complementary results.

Let us start by studying the notion of associativity for internal and external tensor structures on morphisms.

**Definition 8.12.** Let $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$ be internal tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$, respectively; let $\rho$ be an internal tensor structure on $R$ (with respect to $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$).

Suppose that $a_1$ and $a_2$ are associativity constraints on $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$, respectively. We say that $\rho$ is *associative* (with respect to $a_1$ and $a_2$) if it satisfies the following additional condition:

(mor-$a$ITS) For every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}_1(S) \times \mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(S)$

$$ \begin{tikzcd} (R_S(A) \otimes_2 R_S(B)) \otimes_2 R_S(C) \arrow[r, "\rho"] & R_S(A \otimes_1 B) \otimes_2 R_S(C) \arrow[r, "\rho"] & R_S((A \otimes_1 B) \otimes_1 C) \arrow[d, "a_2"] \\ R_S(A) \otimes_2 (R_S(B) \otimes_2 R_S(C)) \arrow[r, "\rho"] & R_S(A) \otimes_2 R_S(B \otimes_1 C) \arrow[r, "\rho"] & R_S(A \otimes_1 (B \otimes_1 C)) \arrow[d, "a_1"'] \end{tikzcd} $$

is commutative.

**Definition 8.13.** Let $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$ be external tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$, respectively; let $\rho$ be an external tensor structure on $R$ (with respect to $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$).

Let $a_1$ and $a_2$ be external associativity constraints on $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$, respectively. We say that $\rho$ is *associative* (with respect to $a_1$ and $a_2$) if it satisfies the following additional condition:

(mor-$a$ETS) For every $S_1, S_2, S_3 \in \mathcal{S}$, the diagram of functors $\mathbb{H}_1(S_1) \times \mathbb{H}_1(S_2) \times \mathbb{H}_1(S_3) \to \mathbb{H}_2(S_1 \times S_2 \times S_3)$
$$ \begin{tikzcd} (R_{S_1}(A_1) \boxtimes R_{S_2}(A_2)) \boxtimes R_{S_3}(A_3) \arrow[r, "\rho"] & R_{S_1 \times S_2}(A_1 \boxtimes A_2) \boxtimes R_{S_3}(A_3) \arrow[r, "\rho"] & R_{S_1 \times S_2 \times S_3}((A_1 \boxtimes A_2) \boxtimes A_3) \arrow[d, "a_2"'] \\ R_{S_1}(A_1) \boxtimes (R_{S_2}(A_2) \boxtimes R_{S_3}(A_3)) \arrow[r, "\rho"] & R_{S_1}(A_1) \boxtimes R_{S_2 \times S_2}(A_2 \boxtimes A_3) \arrow[r, "\rho"] & R_{S_1 \times S_2 \times S_3}(A_1 \boxtimes (A_2 \boxtimes A_3)) \arrow[d, "a_1"'] \end{tikzcd} $$

is commutative.
**Lemma 8.14.** For $j = 1,2$ let $(\otimes_j, m_j; a_j)$ be an associative internal tensor structure on $\mathbb{H}_j$, and let $(\boxtimes_j, m_j^e; a_j^e)$ denote the associative external tensor structure obtained from it via Lemma 2.10 and Lemma 3.3. Moreover, let $\rho$ be an internal tensor structure on $R$ (with respect to $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$), and let $\rho^e$ denote the corresponding external tensor structure on $R$ (with respect to $(\boxtimes_1, m_1^e)$ and $(\boxtimes_2, m_2^e)$).

Suppose that $\rho$ is associative (with respect to $a_1$ and $a_2$). Then $\rho^e$ is associative (with respect to $a_1^e$ and $a_2^e$).

*Proof.* Let us check that the external tensor structure $\rho^e$ satisfies condition (mor-aETS): given $S_1, S_2, S_3 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}_1(S_1) \times \mathbb{H}_1(S_2) \times \mathbb{H}_1(S_3) \to \mathbb{H}_2(S_1 \times S_2 \times S_3)$

$$ \begin{tikzcd} (R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2)) \boxtimes_2 R_{S_3}(A_3) \arrow[r, "\rho^e"] & R_{S_{12}}(A_1 \boxtimes_1 A_2) \boxtimes_2 R_{S_3}(A_3) \arrow[r, "\rho^e"] & R_{S_{123}}((A_1 \boxtimes_1 A_2) \boxtimes_1 A_3) \arrow[d, "a_2^e"] \arrow[d, "a_1^e"] \\ R_{S_1}(A_1) \boxtimes_2 (R_{S_2}(A_2) \boxtimes_2 R_{S_3}(A_3)) \arrow[r, "\rho^e"] & R_{S_1}(A_1) \boxtimes_2 R_{S_{23}}(A_2 \boxtimes_1 A_3) \arrow[r, "\rho^e"] & R_{S_{123}}(A_1 \boxtimes_1 (A_2 \boxtimes_1 A_3)) \end{tikzcd} $$

is commutative. To this end, we decompose it as

<figure><img src="image_59.png" /></figure>

where, for notational convenience, we have set $B_i := \text{pr}_i^{123,*} A_i$, $i = 1,2,3$. Here, the six unnamed natural isomorphisms are obtained by combining connection isomorphisms, internal monoidality isomorphisms and $R$-transition isomorphisms along suitable paths; by Lemma 8.3, we know that the definitions do not depend on the chosen paths. Since the central rectangle in the latter diagram is commutative by axiom (mor-aITS), it now suffices to show that the remaining six lateral pieces are commutative as well. This can be achieved by the usual trick; we omit the details. $\square$

**Lemma 8.15.** For $j = 1,2$ let $(\boxtimes_j, m_j; a_j)$ be an associative external tensor structure on $\mathbb{H}_j$, and let $(\otimes_j, m_j^i; a_j^i)$ denote the associative internal tensor structure obtained from it via Lemma 2.11 and Lemma 3.4. Moreover, let $\rho$ be an external tensor structure on $R$ (with respect to $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$), and let $\rho^i$ denote the corresponding internal tensor structure on $R$ (with respect to $(\boxtimes_1, m_1^i)$ and $(\boxtimes_2, m_2^i)$).

Suppose that $\rho$ is associative (with respect to $a_1$ and $a_2$). Then $\rho^i$ is associative (with respect to $a_1^i$ and $a_2^i$).

*Proof.* Let us check that the internal tensor structure $\rho^i$ satisfies condition (mor-aITS): given $S \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}_1(S) \times \mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(S)$

$$ \begin{tikzcd} (R_S(A) \boxtimes_2 R_S(B)) \boxtimes_2 R_S(C) \arrow[r, "\rho^i"] & R_S(A \otimes_1 B) \boxtimes_2 R_S(C) \arrow[r, "\rho^i"] & R_S((A \otimes_1 B) \boxtimes_1 C) \arrow[d, "a_2^i"] \arrow[d, "a_1^i"] \\ R_S(A) \boxtimes_2 (R_S(B) \boxtimes_2 R_S(C)) \arrow[r, "\rho^i"] & R_S(A) \boxtimes_2 R_S(B \otimes_1 C) \arrow[r, "\rho^i"] & R_S(A \otimes_1 (B \boxtimes_1 C)) \arrow[d, "a_1^i"] \arrow[d, "a_2^i"] \end{tikzcd} $$
is commutative. To this end, we decompose it as

<figure><img src="image_60.png" /></figure>

Here, the six unnamed natural isomorphisms are obtained by combining connection isomorphisms, external monoidality isomorphisms and $R$-transition isomorphisms along suitable paths; by Lemma 8.6, we know that the resulting isomorphisms do not depend on the chosen paths. Since the central rectangle in the latter diagram is commutative by axiom (mor-$a$ETS), it now suffices to show that the remaining six lateral pieces are commutative as well. This can be achieved by the usual trick; we omit the details. $\square$

In the same spirit, let us now study the notion of symmetry for internal and external tensor structures on morphisms.

**Definition 8.16.** Let $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$ be internal tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$, respectively; let $\rho$ be an internal tensor structure on $R$ (with respect to $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$).

Suppose that $c_1$ and $c_2$ are commutativity constraints on $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$, respectively. We say that $\rho$ is symmetric (with respect to $c_1$ and $c_2$) if it satisfies the following additional condition:

(mor-$c$ITS) For every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(S)$

$$ \begin{array}{ccc} R_S(A) \otimes_2 R_S(B) & \xrightarrow{\rho} & R_S(A \otimes_1 B) \\ \bigdownarrow_{c_2} & & \bigdownarrow_{c_1} \\ R_S(B) \otimes_2 R_S(A) & \xrightarrow{\rho} & R_S(B \otimes_1 A) \end{array} $$

is commutative.

**Definition 8.17.** Let $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$ be external tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$, respectively; let $\rho$ be an external tensor structure on $R$ (with respect to $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$).

Let $c_1$ and $c_2$ be external commutativity constraints on $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$, respectively. We say that $\rho$ is symmetric (with respect to $c_1$ and $c_2$) if it satisfies the following additional condition:

(mor-$c$ETS) For every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}_1(S_1) \times \mathbb{H}_1(S_2) \to \mathbb{H}_2(S_1 \times S_2)$

$$ \begin{array}{ccc} R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2) & \xrightarrow{\rho} & R_{S_1 \times S_2}(A_1 \boxtimes_1 A_2) \\ \bigdownarrow_{c_2} & & \bigdownarrow_{c_1} \\ \tau^*(R_{S_2}(A_2) \boxtimes_2 R_{S_1}(A_1)) & \xrightarrow{\rho} & \tau^* R_{S_2 \times S_1}(A_2 \boxtimes_1 A_1) \\ & & \xrightarrow{\theta} \\ & & R_{S_1 \times S_2} \tau^*(A_2 \boxtimes_1 A_1) \end{array} $$

is commutative.

**Lemma 8.18.** For $j = 1, 2$ let $(\otimes_j, m_j; c_j)$ be a symmetric internal tensor structure on $\mathbb{H}_j$, and let $(\boxtimes_j, m_j^e; c_j^e)$ denote the symmetric external tensor structure obtained from it via Lemma 2.10 and Lemma 4.3. Moreover, let $\rho$ be an internal tensor structure on $R$ (with respect to $(\otimes_1, m_1)$ and $(\boxtimes_2, m_2)$), and let $\rho^e$ denote the corresponding external tensor structure on $R$ (with respect to $(\boxtimes_1, m_1^e)$ and $(\boxtimes_2, m_2^e)$).
Suppose that $\rho$ is symmetric (with respect to $c_1$ and $c_2$). Then $\rho^e$ is symmetric (with respect to $c_1^e$ and $c_2^e$).

*Proof.* Let us check that the external tensor structure $\rho^e$ satisfies condition (mor-cETS): given $S_1, S_2 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}_1(S_1) \times \mathbb{H}_1(S_2) \to \mathbb{H}_2(S_1 \times S_2)$

$$ \begin{tikzcd} R_{S_1}(A_1) \boxtimes_2 R_{S_2}(A_2) \arrow[r, "\rho^e"] \arrow[d, "c_2^e"] & R_{S_1 \times S_2}(A_1 \boxtimes_1 A_2) \arrow[d, "c_1^e"] \\ \tau^*(R_{S_2}(A_2) \boxtimes_2 R_{S_1}(A_1)) \arrow[r, "\rho^e"] \arrow[d, "c_2^e"] & \tau^*(R_{S_2 \times S_1}(A_2 \boxtimes_1 A_1)) \arrow[r, "\theta"] \arrow[d, "c_1^e"] & R_{S_1 \times S_2} \tau^*(A_2 \boxtimes_1 A_1) \end{tikzcd} $$

is commutative. To this end, we decompose it as

<figure><img src="image_61.png" /></figure>

Here, the four unnamed natural isomorphisms are obtained by combining connection isomorphisms, internal monoidality isomorphisms and $R$-transition isomorphisms along suitable paths; by Lemma 8.3, we know that the definitions do not depend on the chosen paths. Since the central sauqre is commutative by axiom (mor-cITS), it now suffices to show that the remaining four lateral pieces are commutative as well. This can be achieved by the usual trick; we omit the details. $\square$

**Lemma 8.19.** For $j = 1, 2$ let $(\boxtimes_j, m_j; c_j)$ be a symmetric external tensor structure on $\mathbb{H}_j$, and let $(\otimes_j, m_j^i; c_j^i)$ denote the symmetric internal tensor structure obtained from it via Lemma 2.11 and Lemma 4.4. Moreover, let $\rho$ be an external tensor structure on $R$ (with respect to $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$), and let $\rho^i$ denote the corresponding internal tensor structure on $R$ (with respect to $(\otimes_1, m_1^i)$ and $(\otimes_2, m_2^i)$).

Suppose that $\rho$ is symmetric (with respect to $c_1$ and $c_2$). Then $\rho^i$ is associative (with respect to $c_1^i$ and $c_2^i$).

*Proof.* Let us check that the internal tensor structure $\rho^i$ satisfies condition (mor-cITS): given $S \in \mathcal{S}$, we have to show that the outer part of the diagram of functors $\mathbb{H}_1(S) \times \mathbb{H}_1(S) \to \mathbb{H}_2(S)$

$$ \begin{tikzcd} R_S(A) \otimes_2 R_S(B) \arrow[r, "\rho^i"] \arrow[d, "c_2^i"] & R_S(A \otimes_1 B) \arrow[d, "c_1^i"] \\ R_S(B) \otimes_2 R_S(A) \arrow[r, "\rho^i"] \arrow[d, "c_2^i"] & R_S(B \otimes_1 A) \arrow[d, "c_1^i"] \end{tikzcd} $$
is commutative. To this end, we decompose it as

<figure><img src="image_62.png" /></figure>

Here, the four unnamed natural isomorphisms are obtained by combining connection isomorphisms, external monoidality isomorphisms and $R$-transition isomorphisms along suitable paths; by Lemma 8.6, we know that the definitions do not depend on the chosen paths. Since the central rectangle in the latter diagram is commutative by axiom (mor-cETS), it now suffices to show that the remaining four lateral pieces are commutative as well. This can be achieved by the usual trick; we omit the details. □

Finally, let us study the notion of unitarity for internal and external tensor structures on morphisms.

**Definition 8.20.** Let $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$ be internal tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$, respectively; let $\rho$ be an internal tensor structure on $R$ (with respect to $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$).

Let $u_1$ and $u_2$ be internal unit constraints with unit sections $\mathbf{1}^{(1)}$ and $\mathbf{1}^{(2)}$ on $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$, respectively, and let $w : R(\mathbf{1}^{(1)}) \xrightarrow{\sim} \mathbf{1}^{(2)}$ be an isomorphism between section of $\mathbb{H}_2$. We say that $\rho$ is *unitary via w* (with respect to $u_1$ and $u_2$) if it satisfies the following additional condition:

(mor-uITS) For every $S \in \mathcal{S}$, the two diagrams of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(S)$

$$
\begin{tikzcd}
R_S(A) \otimes_2 R_S(\mathbf{1}_S^{(1)}) \arrow[r, "w_S"] & R_S(A) \otimes_2 \mathbf{1}_S^{(2)} \arrow[d, "\rho"] \\
R_S(A \otimes_1 \mathbf{1}_S^{(1)}) \arrow[r, "u_{1,r}"] & R_S(A) \arrow[d, "u_{2,r}] \\
\otimes & \otimes
\end{tikzcd}
$$

$$
\begin{tikzcd}
R_S(\mathbf{1}_S^{(1)}) \otimes_2 R_S(B) \arrow[r, "w_S"] & \mathbf{1}_S^{(2)} \otimes_2 R_S(B) \arrow[d, "\rho"] \\
R_S(\mathbf{1}_S^{(1)} \otimes_1 B) \arrow[r, "u_{1,l}"] & R_S(B) \arrow[d, "u_{2,l}] \\
\otimes & \otimes
\end{tikzcd}
$$

are commutative.

**Definition 8.21.** Let $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$ be external tensor structures on $\mathbb{H}_1$ and $\mathbb{H}_2$, respectively; let $\rho$ be an external tensor structure on $R$ (with respect to $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$).

Let $u_1$ and $u_2$ be external unit constraints with unit sections $\mathbf{1}^{(1)}$ and $\mathbf{1}^{(2)}$ on $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$, respectively, and let $w : R(\mathbf{1}^{(1)}) \xrightarrow{\sim} \mathbf{1}^{(2)}$ be an isomorphism between section of $\mathbb{H}_2$. We say that $\rho$ is *unitary via w* (with respect to $u_1$ and $u_2$) if it satisfies the following additional condition:

(mor-uETS) For every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}_1(S_1) \to \mathbb{H}_2(S_1 \times S_2)$

$$
\begin{tikzcd}
R_{S_1}(A_1) \boxtimes_2 R_{S_2}(\mathbf{1}_{S_2}^{(1)}) \arrow[r, "w"] & R_{S_1}(A_1) \boxtimes_2 \mathbf{1}_{S_2}^{(2)} \arrow[r, "u_r^{(2)}"] & pr_1^{12,*} R_{S_1}(A_1) \arrow[d, "\rho"] \\
R_{S_1 \times S_2}(A_1 \boxtimes_1 \mathbf{1}_{S_2}^{(1)}) \arrow[r, "u_r^{(1)}"] & R_{S_1 \times S_2}(pr_1^{12,*} A_1) \arrow[d, "\theta"] \\
\otimes & \otimes
\end{tikzcd}
$$
and the diagram of functors $\mathbb{H}_1(S_2) \to \mathbb{H}_2(S_1 \times S_2)$

$$
\begin{tikzcd}
R_{S_1}(\mathbf{1}_{S_1}^{(1)}) \boxtimes_2 R_{S_2}(A_2) \arrow[r, "w"] & \mathbf{1}_{S_1}^{(2)} \boxtimes_2 R_{S_2}(A_2) \arrow[r, "u_l^{(2)}"] & pr_2^{12,*} R_{S_2}(A_2) \\
R_{S_1 \times S_2}(\mathbf{1}_{S_1}^{(1)} \boxtimes_1 A_2) \arrow[u, "\rho"] \arrow[r, "u_l^{(1)}"] & & \theta \arrow[u, "\rho"] \\
& R_{S_1 \times S_2}(pr_2^{12,*} A_2) & &
\end{tikzcd}
$$

are commutative.

**Lemma 8.22.** For $j = 1, 2$ let $(\otimes_j, m_j; u_j)$ be a unitary internal tensor structure on $\mathbb{H}_j$ with unit section $\mathbf{1}^{(j)}$, and let $(\boxtimes_j, m_j^e; u_j^e)$ denote the unitary external tensor structure obtained from it via Lemma 2.11 and Lemma 5.3. Moreover, let $\rho$ be an external tensor structure on $R$ (with respect to $(\otimes_1, m_1)$ and $(\otimes_2, m_2))$, and let $\rho^e$ denote the corresponding internal tensor structure on $R$ (with respect to $(\otimes_1, m_1^e)$ and $(\otimes_2, m_2^e)$). Suppose that $\rho$ is unitary via $w$ (with respect to $u_1$ and $u_2$). Then $\rho^e$ is unitary via $w$ (with respect to $u_1^e$ and $u_2^e$).

*Proof.* Let us check that the external tensor structure $\rho^e$ satisfies condition (mor-*uETS): given $S_1, S_2 \in \mathcal{S}$, we have to show that the diagram of functors $\mathbb{H}_1(S_1) \to \mathbb{H}_2(S_1 \times S_2)$

$$
\begin{tikzcd}
R_{S_1}(A_1) \boxtimes_2 R_{S_2}(\mathbf{1}_{S_2}^{(1)}) \arrow[r, "w"] & R_{S_1}(A_1) \boxtimes_2 \mathbf{1}_{S_2}^{(2)} \arrow[r, "u_{2,r}^{e}"] & pr_1^{12,*} R_{S_1}(A_1) \\
R_{S_1 \times S_2}(A_1 \boxtimes_1 \mathbf{1}_{S_2}^{(1)}) \arrow[u, "\rho^e"] \arrow[r, "u_{1,r}^{e}"] & & \theta \arrow[u, "\rho^e"] \\
& R_{S_1 \times S_2}(pr_1^{12,*} A_1) & &
\end{tikzcd}
$$

and the diagram of functors $\mathbb{H}_1(S_2) \to \mathbb{H}_2(S_1 \times S_2)$

$$
\begin{tikzcd}
R_{S_1}(\mathbf{1}_{S_1}^{(1)}) \boxtimes_2 R_{S_2}(A_2) \arrow[r, "w"] & \mathbf{1}_{S_1}^{(2)} \boxtimes_2 R_{S_2}(A_2) \arrow[r, "u_{2,l}^{e}"] & pr_2^{12,*} R_{S_2}(A_2) \\
R_{S_1 \times S_2}(\mathbf{1}_{S_1}^{(1)} \boxtimes_1 A_2) \arrow[u, "\rho^e"] \arrow[r, "u_{1,l}^{e}"] & & \theta \arrow[u, "\rho^e"] \\
& R_{S_1 \times S_2}(pr_2^{12,*} A_2) & &
\end{tikzcd}
$$

are commutative. We only show the commutativity of the first diagram; the case of the second diagram is completely analogous. Unwinding the various definitions, we obtain the more explicit diagram

$$
\begin{tikzcd}
& pr_1^{12,*} R_{S_1}(A_1) \otimes_2 pr_2^{12,*} R_{S_2}(\mathbf{1}_{S_2}^{(1)}) \arrow[r, "w"] & pr_1^{12,*} R_{S_1}(A_1) \otimes_2 pr_2^{12,*} \mathbf{1}_{S_2}^{(2)} \arrow[r, "\mathbf{1}^{(2),*}"] & pr_1^{12,*} R_{S_1}(A_1) \otimes_2 \mathbf{1}_{S_1 \times S_2}^{(2)} \\
& R_{S_1 \times S_2}(pr_1^{12,*} A_1) \otimes_2 R_{S_1 \times S_2}(pr_2^{12,*} \mathbf{1}_{S_2}^{(1)}) \arrow[u, "\theta"] \arrow[r, "u_r^{(2)}"] & & pr_1^{12,*} R_{S_1}(A_1) \\
& R_{S_1 \times S_2}(pr_1^{12,*} A_1 \otimes_1 pr_2^{12,*} \mathbf{1}_{S_2}^{(1)}) \arrow[u, "\rho"] \arrow[r, "\mathbf{1}^{(1),*}"] & R_{S_1 \times S_2}(pr_1^{12,*} A_1 \otimes_1 \mathbf{1}_{S_1 \times S_2}^{(1)}) \arrow[u, "\theta"] \arrow[r, "u_r^{(1)}"] & R_{S_1 \times S_2}(pr_1^{12,*} A_1)
\end{tikzcd}
$$
Remark 8.24. The considerations collected in Remark 8.8 admit obvious refinements to the subclasses of associative or symmetric or unitary tensor structures, both in the internal and in the external setting.

## 9. REMARKS ON THE TRIANGULATED CASE

In this final section, we consider the case of monoidal triangulated fibered categories in more detail. Our main goal is to explain how the various notions collected in the previous sections should be adapted in the case of triangulated fibered categories in order for them to interact well with the triangulated structure; in fact, the main point is to spell out their compatibility with the shifting functors.

As usual, we work over a fixed base category $\mathcal{S}$; when discussing (external) tensor structures on $\mathcal{S}$-fibered categories, we will ask $\mathcal{S}$ to admit binary products.

**9.1. Triangulated fibered categories.** In the first place, we complete the general preliminaries of Section 1 by reviewing triangulated $\mathcal{S}$-fibered categories and morphisms of such. For notational convenience, whenever considering a triangulated functor, we write the associated compatibility isomorphisms with shifting functors as mere equalities. In order to avoid confusion, until the end of the present subsection we indicate all connection isomorphisms of $\mathcal{S}$-fibered categories explicitly (rather than also as equalities as stated in Notation 2.1).

**Definition 9.1.** A triangulated $\mathcal{S}$-fibered category $\mathcal{H}$ is the datum of

* for every $S \in \mathcal{S}$, a triangulated category $\mathcal{H}(S)$,

* for every morphism $f: T \to S$ in $\mathcal{S}$, a triangulated functor

$$f^*: \mathcal{H}(S) \to \mathcal{H}(T),$$

called the *inverse image functor* along $f$,

* for every pair of composable morphisms $f: T \to S$ and $g: S \to V$ in $\mathcal{S}$, a natural isomorphism of functors $\mathcal{H}(V) \to \mathcal{H}(T)$

$$\mathrm{conn} = \mathrm{conn}_{f,g} : (gf)^*A \xrightarrow{\sim} f^*g^*A,$$

called the *connection isomorphism* at $(f,g)$

satisfying conditions $(\mathcal{S}$-fib-0) and $(\mathcal{S}$-fib-1) of Definition 1.1 plus the following additional condition:

(tri$\mathcal{S}$-fib) For every pair of composable morphisms $f: T \to S$, $g: S \to V$ in $\mathcal{S}$, the diagram of functors $\mathcal{H}(V) \to \mathcal{H}(T)$

$$\begin{tikzcd}
(gf)^*(A[1]) \arrow[r, "\cong"] \arrow[d, "\mathrm{conn}_{f,g}"] & ((gf)^*A)[1] \arrow[d, "\mathrm{conn}_{f,g}"] \\
f^*g^*(A[1]) \arrow[r, "\cong"] & f^*((g^*A)[1]) \arrow[r, "\cong"] & (f^*g^*A)[1]
\end{tikzcd}$$

is commutative.

**Definition 9.2.** Let $\mathbb{H}_1$ and $\mathbb{H}_2$ be triangulated $\mathcal{S}$-fibered categories in the sense of Definition 9.1.

(1) A morphism of triangulated $\mathcal{S}$-fibered categories $R: \mathbb{H}_1 \to \mathbb{H}_2$ is the datum of

* for every $S \in \mathcal{S}$, a triangulated functor

$$R_S: \mathbb{H}_1(S) \to \mathbb{H}_2(S),$$

* for every morphism $f: T \to S$ in $\mathcal{S}$, a natural isomorphism of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(T)$

$$\theta = \theta_f : f^*R_S(A) \xrightarrow{\sim} R_T(f^*A),$$

called the *R-transition isomorphism* along $f$

satisfying condition (mor-$\mathcal{S}$-fib) of Definition 1.6(1) plus the following additional condition:
(mor-tri$\mathcal{S}$-fib) For every morphism $f: T \to S$ in $\mathcal{S}$, the diagram of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(T)$

$$ \begin{array}{ccc} f^*R_S(A[1]) & = & f^*(R_S(A)[1]) = (f^*R_S(A))[1] \\ \bigdownarrow_{\theta_f} & & \bigdownarrow_{\theta_f} \\ R_T(f^*(A[1])) & = & R_T((f^*A)[1]) = (R_T(f^*A))[1] \end{array} $$

is commutative.

(2) If we are already given a family of triangulated functors $\mathcal{R} := \{R_S : \mathbb{H}_1(S) \to \mathbb{H}_2(S)\}_{S \in \mathcal{S}}$, we say that an $\mathcal{S}$-structure on $\mathcal{R}$ in the sense of Definition 1.1(2) is *triangulated* if the resulting morphism of $\mathcal{S}$-fibered categories $R : \mathbb{H}_1 \to \mathbb{H}_2$ satisfies condition (mor-tri$\mathcal{S}$-fib).

**9.2. Triangulated tensor structures.** From now on, we assume that $\mathcal{S}$ admits binary products. Throughout this section, we consider one fixed $\mathcal{S}$-fibered category $\mathbb{H}$, which we assume to be triangulated in the sense of Definition 9.1.

**Definition 9.3.** (1) A triangulated internal tensor structure $(\otimes, m)$ on $\mathbb{H}$ is the datum of
* for every $S \in \mathcal{S}$, a bi-triangulated functor

$$ - \otimes - = - \otimes_S - : \mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S), $$

called the *internal tensor product functor* over $S$,

* for every morphism $f: T \to S$ in $\mathcal{S}$, a natural isomorphism of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ m = m_f : f^*A \otimes f^*B \xrightarrow{\sim} f^*(A \otimes B), $$

called the *internal monoidality isomorphism* along $f$

satisfying condition (mITS) of Definition 2.2 plus the following two additional conditions:

(triITS-1) For every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{array}{ccc} A[1] \otimes B[1] & \xrightarrow{\sim} & (A \otimes B[1])[1] \\ \bigdownarrow^{\sim} & & \bigdownarrow^{\sim} \\ (A[1] \otimes B)[1] & \xrightarrow{\sim} & (A \otimes B)[2] \end{array} $$

is anti-commutative.

(triITS-2) For every morphism $f: T \to S$ in $\mathcal{S}$, the two diagrams of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(T)$

$$ \begin{array}{ccc} f^*(A[1]) \otimes f^*B & \xrightarrow{m} & f^*(A[1] \otimes B) \\ \parallel & & \parallel \\ (f^*A)[1] \otimes f^*B & & f^*((A \otimes B)[1]) \\ \bigdownarrow^{\sim} & & \bigdownarrow^{\sim} \\ (f^*A \otimes f^*B)[1] & \xrightarrow{m} & f^*(A \otimes B)[1] \end{array} \qquad \begin{array}{ccc} f^*A \otimes f^*(B[1]) & \xrightarrow{m} & f^*(A \otimes B[1]) \\ \parallel & & \parallel \\ f^*A \otimes (f^*B)[1] & & f^*((A \otimes B)[1]) \\ \bigdownarrow^{\sim} & & \bigdownarrow^{\sim} \\ (f^*A \otimes f^*B)[1] & \xrightarrow{m} & f^*(A \otimes B)[1] \end{array} $$

are commutative.

(2) Let $(\otimes, m)$ and $(\otimes', m')$ be two triangulated internal tensor structures on $\mathbb{H}$. An equivalence of internal tensor structures $e: (\otimes, m) \xrightarrow{\sim} (\otimes', m')$ is *triangulated* if it satisfies the following additional condition:
(eq-triITS) For every $S \in \mathcal{S}$, the two diagrams of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{tikzcd} A[1] \otimes B \arrow[r, "e"] \arrow[d, "\zeta"] & A[1] \otimes' B \arrow[d, "\zeta"] \\ (A \otimes B)[1] \arrow[r, "e"] & (A \otimes' B)[1] \arrow[d, "\zeta"] \arrow[d, "\zeta"'] \end{tikzcd} \quad \begin{tikzcd} A \otimes B[1] \arrow[r, "e"] & A \otimes' B[1] \arrow[d, "\zeta"] \\ (A \otimes B)[1] \arrow[r, "e"] & (A \otimes' B)[1] \arrow[d, "\zeta"] \arrow[d, "\zeta"'] \end{tikzcd} $$

are commutative.

*Remark 9.4.* The sign choice in axiom (triITS-1) above agrees with that of [Ayo07a, Defn. 2.1.148]. This choice is motivated by the behavior of the tensor product on the derived category of vector spaces over a field; the same rule holds for the bi-derived functor of any bi-exact functor between abelian categories.

**Definition 9.5.** (1) A triangulated external tensor structure $(\boxtimes, m)$ on $\mathbb{H}$ is the datum of

* for every $S_1, S_2 \in \mathcal{S}$, a bi-triangulated functor

$$ - \boxtimes - = - \boxtimes_S - : \mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2), $$

called the *external tensor product functor* at $(S_1, S_2)$,

* for every choice of morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, $i = 1, 2$, a natural isomorphism of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(T_1 \times T_2)$

$$ m = m_{f_1, f_2} : f_1^* A_1 \boxtimes f_2^* A_2 \xrightarrow{\sim} (f_1 \times f_2)^*(A_1 \boxtimes A_2), $$

called the *external monoidality isomorphism* along $(f_1, f_2)$

satisfying condition (mETS) of Definition 2.6 plus the following two additional conditions:

(triETS-1) For every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_2) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{tikzcd} A_1[1] \boxtimes A_2[1] \arrow[r, "\sim"] \arrow[d, "\zeta"] & (A_1 \boxtimes A_2[1])[1] \arrow[d, "\zeta"] \\ (A_1[1] \boxtimes A_2)[1] \arrow[r, "\sim"] & (A_1 \boxtimes A_2)[2] \arrow[d, "\zeta"'] \arrow[d, "\zeta"'] \end{tikzcd} $$

is anti-commutative.

(triETS-2) For every two morphisms $f_i : T_i \to S_i$ in $\mathcal{S}$, $i = 1, 2$, the two diagrams of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(T_1 \times T_2)$

$$ \begin{tikzcd} f_1^*(A_1[1]) \boxtimes f_2^* A_2 \arrow[r, "m"] \arrow[d, "\parallel"] & (f_1 \times f_2)^*(A_1[1] \boxtimes A_2) \arrow[d, "\zeta"] \\ (f_1^*(A_1)[1] \boxtimes f_2^* A_2) \arrow[r, "m"] \arrow[d, "\parallel"] & (f_1 \times f_2)^*((A_1 \boxtimes A_2)[1]) \arrow[d, "\zeta"] \\ (f_1^*(A_1 \boxtimes f_2^* A_2)[1] \arrow[r, "m"] \arrow[d, "\parallel"] & (f_1 \times f_2)^*(A_1 \boxtimes A_2)[1] \arrow[d, "\zeta"] \arrow[d, "\zeta"'] \end{tikzcd} \quad \begin{tikzcd} f_1^* A_1 \boxtimes f_2^*(A_2[1]) \arrow[r, "m"] \arrow[d, "\parallel"] & (f_1 \times f_2)^*(A \boxtimes B[1]) \arrow[d, "\zeta"] \\ f_1^* A_1 \boxtimes (f_2^* A_2)[1] \arrow[r, "m"] \arrow[d, "\parallel"] & (f_1 \times f_2)^*((A_1 \boxtimes A_2)[1]) \arrow[d, "\zeta"] \\ (f_1^* A_1 \boxtimes f_2^* A_2)[1] \arrow[r, "m"] \arrow[d, "\parallel"] & (f_1 \times f_2)^*(A_1 \boxtimes A_2)[1] \arrow[d, "\zeta"] \arrow[d, "\zeta"'] \end{tikzcd} $$

are commutative.

(2) Let $(\boxtimes', m')$ and $(\boxtimes, m)$ be two triangulated external tensor structures on $\mathbb{H}$. An equivalence of external tensor structures $e : (\boxtimes', m') \xrightarrow{\sim} (\boxtimes, m)$ is *triangulated* if it satisfies the following additional condition:

(eq-triETS) For every $S_1, S_2 \in \mathcal{S}$, the two diagrams of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{tikzcd} A_1[1] \boxtimes' A_2 \arrow[r, "e"] \arrow[d, "\zeta"] & A_1[1] \boxtimes A_2 \arrow[d, "\zeta"'] \\ (A_1 \boxtimes' A_2)[1] \arrow[r, "e"] & (A_1 \boxtimes A_2)[1] \arrow[d, "\zeta"'] \arrow[d, "\zeta"'] \end{tikzcd} \quad \begin{tikzcd} A_1 \boxtimes' A_2[1] \arrow[r, "e"] \arrow[d, "\zeta"] & A_1 \boxtimes A_2[1] \arrow[d, "\zeta"'] \\ (A_1 \boxtimes' A_2)[1] \arrow[r, "e"] & (A_1 \boxtimes A_2)[1] \arrow[d, "\zeta"'] \arrow[d, "\zeta"'] \end{tikzcd} $$
are commutative.

We can then state the triangulated analogue of Proposition 2.12 as follows:

**Proposition 9.6.** The constructions of Lemma 2.10 and Lemma 2.11 canonically define mutually inverse bijections between triangulated equivalence classes of triangulated internal and external tensor structures on $\mathbb{H}$.

*Proof.* In the first place, one needs to show that the constructions of Lemma 8.14 and Lemma 8.15 can be canonically promoted to the triangulated setting. Note that there exists an obvious way to make the single external and internal tensor product functors constructed in this way into bi-triangulated functors; it then suffices to check that this respects the additional conditions of Definition 9.3(1) and Definition 9.5(1). In the second place, one has to show that the canonical equivalences constructed in the proof of Proposition 2.12 are triangulated in the sense of Definition 9.3(2) and Definition 9.5(2). Both tasks are easy and left to the interested reader. □

**9.3. Triangulated associativity, commutativity and unit constraints.** We go on by explaining how to adapt associativity, commutativity and unit constraints to the triangulated case, both in the internal and in the external setting.

**Definition 9.7.** Let $(\otimes, m)$ be a triangulated internal tensor structure on $\mathbb{H}$.

(1) A triangulated associativity constraint $a$ on $(\otimes, m)$ is the datum of

• for every $S \in \mathcal{S}$, a tri-triangulated natural transformation of tri-triangulated functors

$$a = a_S : (A \otimes B) \otimes C \xrightarrow{\sim} A \otimes (B \otimes C)$$

satisfying conditions (aITS-1) and (aITS-2) of Definition 3.1 plus the following additional condition (triaITS) For every $S \in \mathcal{S}$, the three diagrams of functors $\mathbb{H}(S) \times \mathbb{H}(S) \times \mathbb{H}(S)$

$$ \begin{tikzcd} (A[1] \otimes B) \otimes C \arrow[r, "\sim"] \arrow[d, "\sim"] & (A \otimes B)[1] \otimes C \arrow[r, "\sim"] \arrow[d, "\sim"] & ((A \otimes B) \otimes C)[1] \arrow[d, "\sim"] \\ A[1] \otimes (B \otimes C) \arrow[r, "\sim"] \arrow[d, "a"] & (A \otimes (B \otimes C))[1] \arrow[d, "a"] & (A \otimes (B \otimes C))[1] \arrow[d, "a"] \\ (A \otimes B[1]) \otimes C \arrow[r, "\sim"] \arrow[d, "a"] & (A \otimes B)[1] \otimes C \arrow[r, "\sim"] \arrow[d, "a"] & ((A \otimes B) \otimes C)[1] \arrow[d, "a"] \\ A \otimes (B[1] \otimes C) \arrow[r, "\sim"] \arrow[d, "a"] & A \otimes (B \otimes C)[1] \arrow[r, "\sim"] \arrow[d, "a"] & (A \otimes (B \otimes C))[1] \arrow[d, "a"] \\ (A \otimes B) \otimes C[1] \arrow[r, "\sim"] & ((A \otimes B) \otimes C)[1] \arrow[d, "a"] & (A \otimes (B \otimes C))[1] \arrow[d, "a"] \\ A \otimes (B \otimes C[1]) \arrow[r, "\sim"] \arrow[d, "a"] & A \otimes (B \otimes C)[1] \arrow[r, "\sim"] \arrow[d, "a"] & (A \otimes (B \otimes C))[1] \arrow[d, "a"] \end{tikzcd} $$

are commutative.

(2) A triangulated commutativity constraint $c$ on $(\otimes, m)$ is the datum of

• for every $S \in \mathcal{S}$, a bi-triangulated natural transformation of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$c = c_S : A \otimes B \xrightarrow{B} \otimes A$$

satisfying conditions (cITS-1) and (cITS-2) of Definition 4.1 plus the following additional condition:
(tricITS) For every $S \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S) \times \mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{array}{ccc} A[1] \otimes B & \xrightarrow{\sim} & (A \otimes B)[1] \\ \downarrow^{c} & & \downarrow^{c} \\ B \otimes A[1] & \xrightarrow{\sim} & (B \otimes A)[1] \end{array} $$

is commutative.

(3) A *triangulated internal unit constraint* $u$ on $(\otimes, m)$ is the datum of

* a section **1** of the *S*-fibered category **H**, called the *unit section*,

* for every $S \in \mathcal{S}$, two triangulated natural isomorphisms of functors $\mathbb{H}(S) \to \mathbb{H}(S)$

$$ u_r = u_{r,S} : A \otimes \mathbf{1}_S \xrightarrow{\sim} A, \quad u_l = u_{l,S} : \mathbf{1}_S \otimes B \xrightarrow{\sim} B $$

satisfying conditions (uITS-1) and (uITS-2) of Definition 5.1 plus the following additional condition:
(triuITS) For every $S \in \mathcal{S}$, the two diagrams of functors $\mathbb{H}(S) \to \mathbb{H}(S)$

$$ \begin{array}{ccc} A[1] \otimes \mathbf{1}_S & \xrightarrow{u_r} & A[1] \\ \downarrow^{\sim} & & \downarrow^{u_r} \\ (A \otimes \mathbf{1}_S)[1] & & (\mathbf{1}_S \otimes B)[1] \end{array} $$

are commutative.

**Definition 9.8.** Let $(\boxtimes, m)$ be a triangulated external tensor structure on $\mathbb{H}$.

(1) A *triangulated external associativity constraint* $a$ on $(\boxtimes, m)$ is the datum of

* for every $S_1, S_2, S_3 \in \mathcal{S}$, a tri-triangulated natural isomorphism of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(S_1 \times S_2 \times S_3)$

$$ a = a_{S_1,S_2,S_3} : (A_1 \boxtimes A_2) \boxtimes A_3 \xrightarrow{\sim} A_1 \boxtimes (A_2 \boxtimes A_3) $$

satisfying conditions (aETS-1) and (aETS-2) of Definition 3.2 plus the following additional condition:
(triaETS) For every $S_1, S_2 \in \mathcal{S}$, the three diagrams of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \times \mathbb{H}(S_3) \to \mathbb{H}(S_1 \times S_2 \times S_3)$

$$ \begin{array}{ccc} (A_1[1] \boxtimes A_2) \boxtimes A_3 & \xrightarrow{\sim} & (A_1 \boxtimes A_2)[1] \boxtimes A_3 \xrightarrow{\sim} ((A_1 \boxtimes A_2) \boxtimes A_3)[1] \\ \downarrow^{a} & & \downarrow^{a} \\ A_1[1] \boxtimes (A_2 \boxtimes A_3) & \xrightarrow{\sim} & (A_1 \boxtimes (A_2 \boxtimes A_3))[1] \\ (A_1 \boxtimes A_2[1]) \boxtimes A_3 & \xrightarrow{\sim} & (A_1 \boxtimes A_2)[1] \boxtimes A_3 \xrightarrow{\sim} ((A_1 \boxtimes A_2) \boxtimes A_3)[1] \\ \downarrow^{a} & & \downarrow^{a} \\ A_1 \boxtimes (A_2[1] \boxtimes A_3) & \xrightarrow{\sim} & A_1 \boxtimes (A_2 \boxtimes A_3)[1] \xrightarrow{\sim} (A_1 \boxtimes (A_2 \boxtimes A_3))[1] \\ (A_1 \boxtimes A_2) \boxtimes A_3[1] & \xrightarrow{\sim} & ((A_1 \boxtimes A_2) \boxtimes A_3)[1] \\ \downarrow^{a} & & \downarrow^{a} \\ A_1 \boxtimes (A_2 \boxtimes A_3[1]) & \xrightarrow{\sim} & A_1 \boxtimes (A_2 \boxtimes A_3)[1] \xrightarrow{\sim} (A_1 \boxtimes (A_2 \boxtimes A_3))[1] \end{array} $$

are commutative.

(2) A *triangulated external commutativity constraint* $c$ on $(\boxtimes, m)$ is the datum of

* for every $S_1, S_2 \in \mathcal{S}$, a bi-triangulated natural isomorphism of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ c = c_{S_1,S_2} : A_1 \boxtimes A_2 \xrightarrow{\sim} \tau^*(A_2 \boxtimes A_1) $$
satisfying conditions (cETS-1) and (cETS-2) of Definition 4.2 plus the following additional condition: (tricETS) For every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_1) \times \mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{tikzcd} A_1[1] \boxtimes A_2 \arrow[r, "\sim"] \arrow[d, "c"] & (A_1 \boxtimes A_2)[1] \arrow[d, "c"'] \\ \tau^*(A_2 \boxtimes A_1[1]) \arrow[r, "\sim"] & \tau^*((A_2 \boxtimes A_1)[1]) \arrow[r, "\sim"] & \tau^*(A_2 \boxtimes A_1)[1] \arrow[d, "c"'] \end{tikzcd} $$

is commutative.

(3) A triangulated external unit constraint $u$ on $(\boxtimes, m)$ is the datum of

• a section $\mathbf{1}$ of the $\mathcal{S}$-fibered category $\mathbb{H}$, called the unit section,

• for every $S_1, S_2 \in \mathcal{S}$, a triangulated natural isomorphism of functors $\mathbb{H}(S_1) \to \mathbb{H}(S_1 \times S_2)$

$$ u_r = u_{r,S_1,S_2} : A_1 \boxtimes \mathbf{1}_{S_2} \xrightarrow{\sim} pr_1^{12,*} A_1 $$

and a natural isomorphism of functors $\mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ u_l = u_{l,S_1,S_2} : \mathbf{1}_{S_1} \boxtimes A_2 \xrightarrow{\sim} pr_2^{12,*} A_2 $$

satisfying conditions (uETS-1) and (uETS-2) of Definition 5.2 plus the following additional condition:

(triuETS) For every $S_1, S_2 \in \mathcal{S}$, the diagram of functors $\mathbb{H}(S_1) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{tikzcd} A_1[1] \boxtimes \mathbf{1}_{S_2} \arrow[r, "u_r"] \arrow[d, "\sim"] & pr_1^{12,*}(A_1[1]) \arrow[d, "\parallel"'] \\ (A_1 \boxtimes \mathbf{1}_{S_2})[1] \arrow[r, "u_r"] & (pr_1^{12,*} A_1)[1] \arrow[d, "\parallel"'] \end{tikzcd} $$

and the diagram of functors $\mathbb{H}(S_2) \to \mathbb{H}(S_1 \times S_2)$

$$ \begin{tikzcd} \mathbf{1}_{S_1} \boxtimes A_2 \arrow[r, "u_l"] \arrow[d, "\sim"] & pr_2^{12,*}(A_2[1]) \arrow[d, "\parallel"'] \\ (\mathbf{1}_{S_1} \boxtimes A_2)[1] \arrow[r, "u_l"] & (pr_2^{12,*} A_2)[1] \arrow[d, "\parallel"'] \end{tikzcd} $$

are commutative.

**Proposition 9.9.** *The following statements hold:*

(1) The constructions of Lemma 3.3 and Lemma 3.4 define mutually inverse bijections between triangulated equivalence classes of triangulated associative internal and external tensor structures on $\mathbb{H}$.

(2) The constructions of Lemma 4.3 and Lemma 4.4 define mutually inverse bijections between triangulated equivalence classes of triangulated symmetric internal and external tensor structures on $\mathbb{H}$.

(3) The constructions of Lemma 5.3 and Lemma 5.4 define mutually inverse bijections between triangulated equivalence classes of triangulated unitary internal and external tensor structures on $\mathbb{H}$.

*Proof.* One has to show that the two constructions mentioned in each statement can be canonically promoted to the triangulated setting. Note that, in each case, there exists an obvious way to make the single natural isomorphisms in the external and internal setting into multi-triangulated natural isomorphisms. Thus one only needs to check that this respects the relevant additional conditions listed in Definition 9.7 and Definition 9.8. This is easy and left to the interested reader. $\square$