adjoint functor

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###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
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> (The special case in [[Cat]] of the general notion of *[[adjunction]]*.)

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The concept of _adjoint functors_ [[Kan (1958)](#Kan58)] is a key concept in [[category theory]] --- if not _the_ key concept --- and it is in large part through the manifold identification of examples of adjoint functors appearing ubiquituously in the practice of [[mathematics]] that category theoretic tools are brought to use in general mathematics.

Abstractly, the notion of adjoint functors embodies the concept of *[[representable functors]]* and has as special cases the fundamental [[universal constructions]] of [[category theory]] such as notably *[[Kan extensions]]* and hence *([[colimit|co-]])[[limits]]* and *([[coend|co-]])[[ends]]*, while being itself the archetypical special case of a natural notion of *[[adjunction]]* which in [[2-category theory]] embodies a general [principle of duality](geometry+of+physics+--+categories+and+toposes#CategoryTheoryIsTheoryOfDuality).

Concretely, the concept of adjoint functors $L \dashv R \,\colon\, \mathcal{D} \leftrightarrows \mathcal{C}$ is immediately transparent and compelling in its incarnation as a [[natural isomorphism]] on [[hom-sets]] (see [below](#InTermsOfHomIsomorphism)) and more generally on [[hom-objects]] (see at *[[enriched adjoint functor]]*), where it just says that adjoint functors are those that may [[coherence condition|coherently]] be switched left$\leftrightarrow$right in a [[hom-set]]
$$
  L \dashv R
    \;\colon\;
  \mathcal{D} \underoverset{R}{L}{\leftrightarrows} \mathcal{C}
  \;\;\;\;\;\;\;\;\;\;\;\;\;
    \text{means}
  \;\;\;\;\;\;\;\;\;\;\;\;\;
  \mathcal{D}\bigl(L(-),\,-\bigr)
  \;\simeq\;
  \mathcal{C}\bigl(-,\,R(-)\bigr)
  \,.
$$
The striking analogy (in fact a kind of [[categorification]]) of this defining relation to the older notion of *[[adjoint operator|adjoint linear operators]]* between [[Hermitian vector spaces]]/[[Hilbert spaces]] $\mathscr{H}_i$ with [[inner products]] $\langle -,-\rangle_i$
$$
  L
    =
  R^\dagger
    \;\colon\;
  \mathscr{H}_2 \leftrightarrows \mathscr{H}_1
  \;\;\;\;\;\;\;\;\;\;\;
    \text{means}
  \;\;\;\;\;\;\;\;\;\;\;
  \bigl\langle
    L(-),\, -
  \bigr\rangle_2
  \;=\;
  \bigl\langle
    -,\, R(-)
  \bigr\rangle_1
$$
is what gives the notion of adjoint functors its name (cf. further discussion at *[adjoint operator -- history](adjoint+operator#History)*).

> "the universality of the concept of adjointness, which was first isolated and named in the conceptual sphere of category theory" [[Lawvere (1969)](#Lawvere69)]

> "The multiple examples, here and elsewhere, of adjoint functors tend to show that adjoints occur almost everywhere in many branches of Mathematics. It is the thesis of [this book](#MacLane71) that a systematic use of all these adjunctions illuminates and clarifies these subjects." [[MacLane (1971), p. 103](#MacLane71)]

> "In all those areas where category theory is actively used the categorical concept of adjoint functor has come to play a key role." [first line from _[An interview with William Lawvere](https://ncatlab.org/nlab/show/William+Lawvere#Interview07)_, paraphrasing the first paragraph of _[Taking categories seriously](William+Lawvere#TakingCategoriesSeriously)_]

## Definition

There are various different but equivalent characterizations of adjoint functors, some of which are discussed below.

### In terms of Hom isomorphism
 {#InTermsOfHomIsomorphism}

We discuss here the definition of adjointness of functors $L \dashv R$ in terms of a [[natural bijection]] between [[hom-sets]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets} below):

$$
  \{L(c) \to d\} \;\simeq\; \{ c \to R(d) \}
$$

We show that this is equivalent to the abstract definition, in terms of an [[adjunction]] in the [[2-category]] [[Cat]], in Prop. \ref{AdjointnessInTermsOfHomIsomorphismEquivalentToAdjunctionInCat} below.

$\,$

+-- {: .num_defn #AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}
###### Definition
**([[adjoint functors]] in terms of [[natural bijections]] of [[hom-sets]])**

Let $\mathcal{C}$ and $\mathcal{D}$ be two [[categories]], and let

$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{}
  \mathcal{C}
$$

be a pair of [[functors]] between them, as shown. Then this is called a _pair of [[adjoint functors]]_ (or an _[[adjoint pair]] of [[functors]]_) with $L$ _[[left adjoint]]_ and $R$ _[[right adjoint]]_, denoted

\begin{center}
  \begin{tikzcd}
    \mathcal{D}
     \arrow[r, shift right=6pt, "R"', "\bot"]
    &
    \mathcal{C}
     \arrow[l, shift right=6pt, "L"']
  \end{tikzcd}
\end{center}

if there exists a [[natural isomorphism]] between the [[hom-functors]] of the following form:

\[
  \label{HomIsomorphismForAdjointFunctors}
  Hom_{\mathcal{D}}\big(L(-),\,-\big)
   \;\simeq\;
  Hom_{\mathcal{C}}\big(-,\,R(-)\big)
  \,.
\]

This means that for all [[objects]] $c \in \mathcal{C}$ and $d \in \mathcal{D}$ there is a [[bijection]] of [[hom-sets]]

$$
  \array{
    Hom_{\mathcal{D}}(L(c),d)
      &\overset{\simeq}{\longrightarrow}&
    Hom_{\mathcal{C}}(c,R(d))
    \\
    ( L(c) \overset{f}{\to} d )
    &\mapsto&
    (c \overset{\widetilde f}{\to} R(d))
  }
$$

which is [[natural bijection|natural]] in $c$ and $d$.  This isomorphism is the **adjunction isomorphism** and the [[image]] $\widetilde f$ of a morphism $f$ under this bijections is called the _[[adjunct]]_ of $f$. Conversely, $f$ is called the adjunct of $\widetilde f$.

Naturality here means that for every [[morphism]] $g \colon c_2 \to c_1$ in $\mathcal{C}$ and for every [[morphism]] $h\colon d_1\to d_2$ in $\mathcal{D}$, the resulting square

\[
  \label{NaturalitySquareForAdjointnessOfFunctors}
  \array{
    Hom_{\mathcal{D}}(L(c_1), d_1)
      &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
    Hom_{\mathcal{C}}(c_1, R(d_1))
    \\
    {}^{\mathllap{Hom_{\mathcal{D}}(L(g), h)}}\big\downarrow
     &&
    \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(g, R(h))}}
    \\
    Hom_{\mathcal{D}}(L(c_2),d_2)
    &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
    Hom_{\mathcal{C}}(c_2,R(d_2))
  }
\]

[[commuting square|commutes]] (see also at _[[hom-functor]]_ for the definition of the vertical maps here).

Explicitly, this commutativity, in turn, means that for every morphism $f \;\colon\; L(c_1) \to d_1$ with [[adjunct]] $\widetilde f \;\colon\; c_1 \to R(d_1)$, the adjunct of the [[composition]] is

$$
  \array{
    L(c_1) & \overset{f}{\longrightarrow} & d_1
    \\
    {}^{\mathllap{L(g)}}\big\uparrow && \big\downarrow^{\mathrlap{h}}
    \\
    L(c_2) && d_2
  }
  \;\;\;=\;\;\;
  \array{
    c_1 &\overset{\widetilde f}{\longrightarrow}& R(d_1)
    \\
    {}^{\mathllap{g}}\big\uparrow && \big\downarrow^{\mathrlap{R(h)}}
    \\
    c_2 && R(d_2)
  }
$$

=--

+-- {: .num_defn #AdjunctionUnitFromHomIsomorphism}
###### Definition
**([[adjunction unit]] and [[adjunction counit|counit]] in terms of hom-isomorphism)

Given a pair of [[adjoint functors]]

$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot}
  \mathcal{C}
$$
according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets} one says that

1. for any $c \in \mathcal{C}$ the [[adjunct]] of the [[identity morphism]] on  $L(c)$ is the _[[unit of an adjunction|unit morphism]]_ of the adjunction at that object, denoted

   $$
     \eta_c \coloneqq \widetilde{id_{L(c)}} \;\colon\; c \longrightarrow R(L(c))
   $$

1. for any $d \in \mathcal{D}$ the [[adjunct]] of the [[identity morphism]] on  $R(d)$ is the _[[counit of an adjunction|counit morphism]]_ of the adjunction at that object, denoted

   $$
     \epsilon_d \;\colon\; L(R(d)) \longrightarrow d
   $$

=--

+-- {: .num_prop #GeneralAdjunctsInTermsOfAdjunctionUnitCounit}
###### Proposition
**(general [[adjuncts]] in terms of [[adjunction unit|unit/counit]])**

Consider a pair of [[adjoint functors]]

$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot}
  \mathcal{C}
$$

according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}, with [[adjunction units]] $\eta_c$ and [[adjunction counits]] $\epsilon_d$ according to Def. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit}.

Then

1. The [[adjunct]] $\widetilde f$ of any morphism $L(c) \overset{f}{\to} d$ is obtained from $R$ and $\eta_c$ as the [[composition|composite]]

   \[
     \label{AdjunctFormula}
     \widetilde f
     \;\colon\;
     c
       \overset{\eta_c}{\longrightarrow}
     R(L(c))
       \overset{R(f)}{\longrightarrow}
     R(d)
   \]

   Conversely, the [[adjunct]] $f$ of any morphism $c \overset{\widetilde f}{\longrightarrow} R(d)$ is obtained from $L$ and $\epsilon_d$ as

   \[
     \label{ConverseAdjunctFormula}
     f
     \;\colon\;
     L(c)
       \overset{L(\widetilde f)}{\longrightarrow}
     L(R(d))
       \overset{\epsilon_d}{\longrightarrow}
     d
   \]

1. The [[adjunction units]] $\eta_c$ and [[adjunction counits]] $\epsilon_d$ are components of [[natural transformations]] of the form

   $$
     \eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L
   $$

   and

   $$
     \epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}
   $$

1. The [[adjunction unit]] and [[adjunction counit]] satisfy the [[triangle identities]], saying that

   $$
      id_{L(c)}
      \;\colon\;
      L(c)
        \overset{L(\eta_c)}{\longrightarrow}
      L(R(L(c)))
        \overset{\epsilon_{L(c)}}{\longrightarrow}
      L(c)
   $$

   and

   $$
     id_{R(d)}
     \;\colon\;
     R(d)
       \overset{\eta_{R(d)}}{\longrightarrow}
     R(L(R(d)))
       \overset{R(\epsilon_d)}{\longrightarrow}
     R(d)
   $$

=--

+-- {: .proof}
###### Proof

For the first statement, consider the [[naturality square]] (eq:NaturalitySquareForAdjointnessOfFunctors) in the form

$$
  \array{
    id_{L(c)}
     \in
    &
    Hom_{\mathcal{D}}(L(c), L(c))
      &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
    Hom_{\mathcal{C}}(c, R(L(c)))
    \\
    &
    {}^{\mathllap{Hom_{\mathcal{D}}(L(id), f)}}\big\downarrow
     &&
    \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(id, R(f))}}
    \\
    &
    Hom_{\mathcal{D}}(L(c), d)
    &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
    Hom_{\mathcal{C}}( c, R(d) )
  }
$$

and consider the element $id_{L(c_1)}$ in the top left entry. Its image under going down and then right in the diagram is $\widetilde f$, by Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}. On the other hand, its image under going right and then down is $ R(f)\circ \eta_{c}$, by Def. \ref{AdjunctionUnitFromHomIsomorphism}. Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown, for the adjunct of $f$.

The converse formula follows analogously.

The third statement follows directly from this by applying these formulas for the [[adjuncts]] twice and using that the result must be the original morphism:

$$
  \begin{aligned}
    id_{L(c)}
    & =
    \widetilde \widetilde { id_{L(c)} }
    \\
    & = \widetilde{ c \overset{\eta_c}{\to} R(L(c))  }
    \\
    & =
     L(c)
       \overset{L(\eta_c)}{\longrightarrow}
     L(R(L(c)))
       \overset{\epsilon_{L(c)}}{\longrightarrow}
     L(c)
  \end{aligned}
$$

For the second statement, we have to show that for every morphism $f \colon c_1 \to c_2$ the following [[commuting square|square commutes]]:

$$
  \array{
     c_1 &\overset{f}{\longrightarrow}& c_2
     \\
     {}^{\mathllap{\eta_{c_1}}}\big\downarrow
       &&
     \big\downarrow^{\mathrlap{\eta_{c_2}}}
     \\
     R(L(c_1))
      &\underset{ R(L(f)) }{\longrightarrow}&
     R(L(c_2))
  }
$$

To see this, consider the [[naturality square]] (eq:NaturalitySquareForAdjointnessOfFunctors) in the form

$$
  \array{
    id_{L(c_2)}
    \in
    & Hom_{\mathcal{D}}(L(c_2), L(c_2))
      &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
    Hom_{\mathcal{C}}(c_2, R(L(c_2)))
    \\
    &
    {}^{\mathllap{Hom_{\mathcal{D}}(L(f),id_{L(c_2)})}}\big\downarrow
     &&
    \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(f, R(id_{L(c_2)}))}}
    \\
    &
    Hom_{\mathcal{D}}(L(c_1),L(c_2))
      &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
    Hom_{\mathcal{C}}(c_1,R(L(c_2)))
  }
$$

The image of the element $id_{L(c_2)}$ in the top left along the right and down is $ \eta_{c_2} \circ f$, by Def. \ref{AdjunctionUnitFromHomIsomorphism}, while its image down and then to the right is $\widetilde{L(f)} = R(L(f)) \circ \eta_{c_1}$, by the previous statement. Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown.

The argument for the naturality of $\epsilon$ is directly analogous.

=--

+-- {: .num_prop #AdjointnessInTermsOfHomIsomorphismEquivalentToAdjunctionInCat}
###### Proposition
**(adjointness in terms of hom-isomorphism equivalent to adjunction in $Cat$)**

Two functors

$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{}
  \mathcal{C}
$$

are an [[adjoint pair]] in the sense that there is a [[natural isomorphism]] (eq:HomIsomorphismForAdjointFunctors) according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}, precisely if they participate in an [[adjunction]] in the [[2-category]] [[Cat]], meaning that

1. there exist [[natural transformations]]

   $$
     \eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L
   $$

   and

   $$
     \epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}
   $$

2. which satisfy the [[triangle identities]]

   $$
      id_{L(c)}
      \;\colon\;
      L(c)
        \overset{L(\eta_c)}{\longrightarrow}
      L(R(L(c)))
        \overset{\epsilon_{L(c)}}{\longrightarrow}
      L(c)
   $$

   and

   $$
     id_{R(d)}
     \;\colon\;
     R(d)
       \overset{\eta_{R(d)}}{\longrightarrow}
     R(L(R(d)))
       \overset{R(\epsilon_d)}{\longrightarrow}
     R(d)
   $$

=--

+-- {: .proof}
###### Proof

That a hom-isomorphism (eq:HomIsomorphismForAdjointFunctors) implies units/counits satisfying the [[triangle identities]] is the statement of the second two items of Prop. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit}.

Hence it remains to show the converse. But the argument is along the same lines as the proof of Prop. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit}: We now _define_ forming of adjuncts by the formula (eq:AdjunctFormula). That the resulting assignment $f \mapsto \widetilde f$ is an [[isomorphism]] follows from the computation

$$
  \begin{aligned}
    \widetilde {\widetilde f}
    & =
    \widetilde{ c \overset{\eta_c}{\to} R(L(c)) \overset{R(f)}{\to} R(d) }
    \\
    & =
    L(c) \overset{L(\eta_c)}{\to} L(R(L(c))) \overset{L(R(f))}{\to} L(R(d)) \overset{\epsilon_d}{\to} d
    \\
    & =
    L(c) \overset{L(\eta_c)}{\to} L(R(L(c)))
         \overset{ \epsilon_{L(c)} }{\to}  L(c)
         \overset{f}{\longrightarrow} d
    \\
    & =  L(c) \overset{f}{\longrightarrow} d
  \end{aligned}
$$

where, after expanding out the definition, we used [[natural transformation|naturality]] of $\epsilon$ and then the [[triangle identity]].

Finally, that this construction satisfies the naturality condition (eq:NaturalitySquareForAdjointnessOfFunctors) follows from the functoriality of the functors involved, and the naturality of the unit/counit:

$$
  \array{
    c_2 &\overset{ \eta_{c_2} }{\longrightarrow}& R(L(c_2))
    \\
    {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{R(L(g))}}
    & \searrow^{\mathrlap{ R( L(g) \circ f ) }}
    \\
    c_1
      &\overset{\eta_{c_1}}{\longrightarrow}&
    R(L(c_1))
     &\overset{R(f)}{\longrightarrow}&
    R(d_1)
    \\
    && & {}_{R( h\circ f)}\searrow & \downarrow^{\mathrlap{ R(h) }}
    \\
    && && R(d_2)
  }
$$

=--

### In terms of representable functors
 {#InTermsOfRepresentableFunctors}

The condition (eq:HomIsomorphismForAdjointFunctors) on adjoint functors $L \dashv R$ in Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets} implies in particular that for every [[object]] $d \in \mathcal{D}$ the functor $Hom_{\mathcal{D}}(L(-),d)$ is a _[[representable functor]]_ with _[[representing object]]_ $R(d)$. The following Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} observes that the existence of such [[representing objects]] for all $d$ is, in fact, already sufficient to imply that there is a right adjoint functor.

This equivalent perspective on adjoint functors makes manifest that:

1. adjoint functors are, if they exist, unique up to natural isomorphism, this is Prop. \ref{UniquenessOfAdjoints} below;

1. the concept of adjoint functors makes sense also _[[relative adjoint functor|relative]]_ to a [[full subcategory]] on which representing objects exists, this is the content of Remark \ref{RelativeAdjointFunctors} below.

#### Global definition

+-- {: .num_prop #AdjointFunctorFromObjectwiseRepresentingObject}
###### Proposition
**(adjoint functor from objectwise [[representing object]])**

A [[functor]] $L \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ has a [[right adjoint]] $R \;\colon\; \mathcal{D} \to \mathcal{C}$, according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}, already if
for all [[objects]] $d \in \mathcal{D}$ there is an object $R(d) \in \mathcal{C}$ such that there is a [[natural isomorphism]]

   $$
     Hom_{\mathcal{D}}(L(-),d)
     \underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}
     Hom_{\mathcal{C}}(-,R(d))
     \,,
   $$

   hence for each [[object]] $c \in \mathcal{C}$ a [[bijection]]

   $$
     Hom_{\mathcal{D}}(L(c),d)
     \underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}
     Hom_{\mathcal{C}}(c,R(d))
   $$

   such that for each [[morphism]] $g \;\colon\; c_2 \to c_1$, the following [[commuting diagram|diagram commutes]]

   \[
     \label{HalfNaturalitySquareForAdjointnessOfFunctors}
     \array{
       Hom_{\mathcal{D}}(L(c_1),d)
         &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
       Hom_{\mathcal{C}}(c_1,R(d))
       \\
       {}^{\mathllap{ Hom_{\mathcal{C}}(L(g),id_d) }}
       \big\downarrow
       &&
       \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( f, id_{R(d)}  ) }}
       \\
       Hom_{\mathcal{D}}(L(c_2),d)
         &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
       Hom_{\mathcal{C}}(c_2,R(d))
     }
   \]

   (This is as in (eq:NaturalitySquareForAdjointnessOfFunctors), except that only naturality in the first variable is required.)

In this case there is a unique way to extend $R$ from a function on [[objects]] to a function on [[morphisms]] such as to make it a [[functor]] $R \colon \mathcal{D} \to \mathcal{C}$ which is [[right adjoint]] to $L$.
, and hence the statement is that with this, naturality in the second variable is already implied.

=--

+-- {: .proof}
###### Proof

Notice that

1. in the language of [[presheaves]] the assumption is that for each $d \in \mathcal{D}$ the presheaf

   $$
     Hom_{\mathcal{D}}(L(-),d)
     \;\in\;
     [\mathcal{C}^{op}, Set]
   $$

   is [[representable functor|represented]] by the object $R(d)$, and [[natural transformation|naturally]] so.

1. In terms of the [[Yoneda embedding]]

   $$
     y
       \;\colon\;
     \mathcal{C}
      \hookrightarrow
     [\mathcal{C}^{op}, Set]
   $$

   we have

   \[
     \label{YonedanotationForRepresentable}
     Hom_{\mathcal{C}}(-,R(d))
     =
     y(R(d))
   \]

The condition (eq:NaturalitySquareForAdjointnessOfFunctors) says equivalently that $R$ has to be such that for all [[morphisms]] $h \;\colon\; d_1 \to d_2 $ the following diagram in the [[category of presheaves]] $[\mathcal{C}^{op}, Set]$ [[commuting diagram|commutes]]

$$
  \array{
    Hom_{\mathcal{D}}(L(-),d_1)
      &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
    Hom_{\mathcal{C}}(-,R(d_1))
    \\
    {}^{\mathllap{ Hom_{\mathcal{C}}( L(-) , h ) }}
    \big\downarrow
    &&
    \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( -, R(h)  ) }}
    \\
    Hom_{\mathcal{D}}(L(-),d_2)
      &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
    Hom_{\mathcal{C}}(-, R(d_2))
  }
$$

This manifestly has a unique solution

$$
  y(R(h))
  \;=\;
  Hom_{\mathcal{C}}(-,R(h))
$$

for every morphism $h \colon d_1 \to d_2$ under  $y(R(-))$ (eq:YonedanotationForRepresentable). But the [[Yoneda embedding]] $y$ is a [[fully faithful functor]] ([this prop.](Yoneda+embedding#YonedaEmbeddingIsFullyFaithful)), which means that thereby also $R(h)$ is uniquely fixed.

=--

+-- {: .num_remark}
###### Remark

In more fancy language, the statement of Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} is the following:

By precomposition $L$ defines a functor of [[presheaf categories]]

$$
  L^* \;\colon\; [\mathcal{D}^{op}, Set] \to [\mathcal{C}^{op}, Set]
  \,.
$$

By [[restriction]] along the [[Yoneda embedding]] $y \;\colon\; \mathcal{D} \to [\mathcal{D}^{op}, Set]$ this yields the functor

$$
  \bar L
  \;\colon\;
  \array{
    \mathcal{D}
      &\overset{y}{\longrightarrow}&
    [\mathcal{D}^{op}, Set]
      &\overset{L^*}{\longrightarrow}&
    [\mathcal{C}^{op}, Set]
    \\
    d
      &\mapsto&
    Hom_{\mathcal{D}}(-,d)
      &\mapsto&
    Hom_{\mathcal{D}}(L(-),d)
 }
 \,.
$$

The statement is that for all $d \in D$ this presheaf $\bar L(d)$ is [[representable functor|representable]], then it is functorially so in that there exists a functor $R \colon \mathcal{D} \to \mathcal{C}$ such that

$$
  \bar L \;\simeq\; y \circ R
  \,.
$$

=--

#### Local definition
 {#LocalDefinition}

+-- {: .num_remark #RelativeAdjointFunctors}
###### Remark
**([[relative adjoint functors]])**

The perspective of Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} has the advantage that it yields useful information even if the adjoint functor $R$ does not exist globally, i.e. as a functor on all of $\mathcal{D}$:

It may happen that

$$
  \bar L(d) \coloneqq Hom_D(L(-),d) \in [C^{op}, Set]
$$

is [[representable functor|representable]] for _some_ [[object]] $d \in \mathcal{D}$ but not for all $d$. The representing object may still usefully be thought of as $R(d)$, and in fact it may be viewed as a right adjoint to $L$ _relative to_ the inclusion of the [[full subcategory]] determined by those $d$s for which $\bar L(d)$ is representable; see [[relative adjoint functor]] for more.

This _global_ versus _local_ evaluation of adjoint functors induces the global/local pictures of the definitions

* [[limit]] / [[homotopy limit]]

* [[Kan extension]]

as discussed there.

=--

### In terms of universal factorization through a (co)unit
 {#UniversalArrows}

We have seen in Prop. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit} that the [[unit of an adjunction]] and [[counit of an adjunction]] plays a special role. One may amplify this by characterizing these morphisms as _[[universal arrows]]_ in the sense of the following Def. \ref{UniversalArrow}. In fact the existence of these is already equivalent to the existence of an adjoint functor, this is the statement of Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor} below.

$\,$

+-- {: .num_defn #UniversalArrow}
###### Definition
**(universal arrow)**

Given a [[functor]] $R \;\colon\; \mathcal{D} \to \mathcal{C}$, and an object $c\in \mathcal{C}$, a _universal arrow_ from $c$ to $R$ is an [[initial object]] of the [[comma category]] $(c/R)$.  This means that it consists of

1. an [[object]] $L(c)\in \mathcal{D}$

1. a [[morphism]] $\eta_c \;\colon\; c \to R\big(L(c)\big)$, to be called the _[[adjunction unit|unit]]_,

such that for any $d\in \mathcal{D}$, any morphism $f \colon c\to R(d)$ factors through this unit $\eta_c$ as

\[
  \label{UniversalArrowFactorization}
  \array{
    && c
    \\
    &
    {}^{\mathllap{\eta_c}}\swarrow
    &&
   \searrow^{\mathrlap{f}}
    \\
    R\big(L(c)\big)
    &&
    \underset
      {R (\widetilde f)}
      {\longrightarrow}
    &&
    R(d)
    \\
    \\
    L(c)
    &&
    \underset{ \widetilde f}{\longrightarrow}
    &&
    d
  }
\]

for a unique $\widetilde f  \;\colon\; L(c) \longrightarrow d$, to be called the [[adjunct]] of $f$.

=--

(e.g. [Borceux, Vol. 1, Definition 3.1.1](#Borceux94))

+-- {: .num_prop #UniversalMorphismsAreInitialObjectsInCommaCategory}
###### Proposition
**([[universal morphisms]] are [[initial objects]] in the [[comma category]])**

Let $R: \mathcal{D} \to \mathcal{C}$ be a [[functor]] and $c \in \mathcal{C}$ an [[object]]. Then the following are equivalent:

1. $c \overset{\eta_c}{\to} R(L(c))$ is a [[universal morphism]] into $R(L(c))$ (Def. \ref{UniversalArrow});

1. $(c, \eta_c)$ is the [[initial object]] in the [[comma category]] $c/R$.

=--

+-- {: .num_prop #CollectionOfUniversalArrowsEquivalentToAdjointFunctor}
###### Proposition
**(collection of [[universal arrows]] equivalent to [[adjoint functor]])**

Let $R \;\colon\; \mathcal{D} \to \mathcal{C}$ be a [[functor]]. Then the following are equivalent:

1. $R$ has a [[left adjoint]] functor $L \colon \mathcal{C} \to \mathcal{D}$ according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets},

1. for every [[object]] $c \in \mathcal{C}$ there is a universal arrow $c \overset{\eta_c}{\longrightarrow} R(L(c))$, according to Def. \ref{UniversalArrow}.

=--

+-- {: .proof}
###### Proof

In one direction, assume a [[left adjoint]] $L$ is given. Define the would-be universal arrow at $c \in \mathcal{C}$ to be the [[unit of an adjunction|unit of the adjunction]] $\eta_c$ via Def. \ref{AdjunctionUnitFromHomIsomorphism}. Then the statement that this really is a universal arrow is implied by Prop. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit}.

In the other direction, assume that universal arrows $\eta_c$ are given. The uniqueness clause in Def. \ref{UniversalArrow} immediately implies  [[bijections]]

$$
  \array{
    Hom_{\mathcal{D}}(L(c),d)
      &\overset{\simeq}{\longrightarrow}&
    Hom_{\mathcal{C}}(c,R(d))
    \\
    \left(
      L(c) \overset{\widetilde f}{\to} d
    \right)
      &\mapsto&
    \left(
      c \overset{\eta_c}{\to} R(L(c)) \overset{ R(\widetilde f) }{\to} R(d)
    \right)
  }
$$

Hence to satisfy (eq:HomIsomorphismForAdjointFunctors) it remains to show that these are [[natural transformation|natural]] in both variables. In fact, by Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} it is sufficient to show naturality in the variable $d$. But this is immediate from the functoriality of $R$ applied in (eq:UniversalArrowFactorization): For $h \colon d_1 \to d_2$ any [[morphism]], we have

$$
  \array{
    && c
    \\
    & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}}
    \\
    R (L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d_1)
    \\
    && {}_{\mathllap{ R( h\circ \widetilde f ) }}\searrow && \downarrow^{\mathrlap{R(h)}}
    \\
    && && R(d_2)
  }
$$

=--

+-- {: .num_example}
###### Example
**([[localization]] via [[universal arrows]])**

The characterization of adjoint functors in terms of universal factorizations through the unit and counit (Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor}) is of particular interest in the case that $R$ is a [[full and faithful functor]]

$$
  R \;\colon\;
  \mathcal{D} \hookrightarrow \mathcal{C}
$$

exhibiting $\mathcal{D}$ as a [[reflective subcategory]] of $\mathcal{C}$. In this case we may think of $L$ as a [[localization]] and of objects in the [[essential image]] of $L$ as **local objects**. Then the above says that:

* every morphism $c \to R d$ from $c$ into a local object factors throught the localization of  $c$.

=--

### In terms of comma categories
 {#InTermsOfCommaCategories}

A [[functor]] $L \colon C \to D$ is [[left adjoint]] to a functor $R \colon D \to C$ if and only if there is an isomorphism (not [[equivalence]]) of [[comma categories]] $L \downarrow D \cong C \downarrow R$ and this isomorphism commutes with the [[forgetful functors]] to the [[product category]] $C \times D$. See §B.I.2 of *[[Functorial Semantics of Algebraic Theories]]*.

This characterisation generalises (in the unenriched setting) to [[relative adjunctions]] by replacing $C \downarrow R$ by $J \downarrow R$.

### In terms of cographs/correspondences/heteromorphisms
 {#InTermsOfCographsHeteromorphisms}

Every [[profunctor]]

$$
  k : C^{op} \times D \to S
$$

defines a category $C *^k D$ with $Obj(C *^k D) = Obj(C) \sqcup Obj(D)$ and with [[hom set]] given by

$$
  Hom_{C^{op} \times D}(X,Y)
  =
  \left\{
    \array{
      Hom_C(X,Y) & if X, Y \in C
      \\
      Hom_{D}(X,Y) & if X,Y \in D
      \\
      k(X,Y) & if X \in C and Y \in D
      \\
      \emptyset & otherwise
    }
  \right.
$$

($k(X,Y)$ is also called the _[[heteromorphisms]]_).

This category naturally comes with a functor to the [[interval]] category

$$
  C *^k D \to \Delta^1
  \,.
$$

Now, every functor $L : C \to D$ induces a [[profunctor]]

$$
  k_L(X,Y) = Hom_D(L(X), Y)
$$

and every functor $R : D \to C$ induces a [[profunctor]]

$$
  k_R(X,Y) = Hom_C(X, R(Y))
  \,.
$$

The functors $L$ and $R$ are adjoint precisely if the [[profunctors]] that they define in the above way are equivalent. This in turn is the case if
$C \star^L D \simeq (D^{op} \star^{R^{op}} C^{op})^{op}$.

We say that $C \star^k D$ is the [[cograph of a functor|cograph of the functor]] $k$. See there for more on this.

### In terms of graphs/2-sided discrete fibrations

Functors $L \colon C \to D$ and $R \colon D \to C$ are adjoint precisely if we have a [[commutative diagram]] of the form

$$
    \array{
       (L \downarrow Id_D)
       &&\stackrel{\cong}{\to}&&
       (Id_C \downarrow R)
       \\
       & \searrow && \swarrow
       \\
       &&
       C \times D
       \mathrlap{\,,}
    }
$$

where the downwards arrows denote the [[maps]] induced by the canonical projections out of the [[comma categories]]. This definition of adjoint functors was introduced by [[Lawvere]] in *[[Functorial Semantics of Algebraic Theories]]*, and was the original motivation for [[comma categories]].

The above [[diagram]] may be recovered directly from the image under the equivalence $[C^{op} \times D, Set] \stackrel{\simeq}{\to} DFib(D,C) $ described at [[2-sided fibration]] of the isomorphism of induced [[profunctors]] $C^{op} \times D \to Set$ (see above at "In terms of Hom isomorphism"). Its relation to the hom-set definition of adjoint functors can thus be understood within the general paradigm of [[Grothendieck construction]]-like correspondences. Consequently, this description is not viable for [[enriched category|enriched adjunctions]].

This description generalises to [[relative adjunctions]] by replacing $Id_C$ with $J$.

### In terms of Kan extensions/liftings

Given $L \colon C \to D$, we have that it has a [[right adjoint]] $R\colon D \to C$ precisely if the [[left Kan extension]] $Lan_L 1_C$ of the [[identity]] along $L$ exists and is [absolute](/nlab/show/Kan+extension#AbsoluteKanExtension), in which case

$$
  R \simeq \mathop{Lan}_L 1_C
  \,.
$$

In this case, the universal 2-cell $1_C \to R L$ corresponds to the
[[unit of an adjunction|unit of the adjunction]]; the counit and the verification of the triangular identities can all be obtained through properties of Kan extensions and absoluteness.

It is also possible to express this in terms of Kan liftings: $L$ has a right adjoint $R$ if and only if:

- $R \simeq \mathop{Rift}_L 1_D$ and this [[Kan lift]] is [[Kan lift|absolute]]

In this case, we get the counit as given by the universal cell $L R \to 1_D$, while the rest of the data and properties can be derived from it through the absolute Kan lifting assumption.

Dually, we have that for $R\colon D \to C$, it has a left adjoint $L \colon C \to D$ precisely if

- $L \simeq \mathop{Ran}_R 1_D$, and this Kan extension is [[Kan extension|absolute]]

or, in terms of left Kan liftings:

- $L \simeq \mathop{Lift}_R 1_C$, and this Kan lifting is [[Kan lift|absolute]]

This follows from the fact that the adjunction $L \dashv R$ [induces](#PrePostcompositeAdjunctions) adjunctions $- \circ R \dashv - \circ L$ and $L \circ - \dashv R \circ -$.

The formulations in terms of liftings generalize to (unenriched) [[relative adjoint|relative adjoints]] by allowing an arbitrary functor $J$ in place of the identity; see there for more.

### Transformation of adjoints
 {#TransformationOfAdjoints}

There are several layers of generality at which one may consider a notion of [[homomorphism]] *between* adjoint functors.

Here is a basic but important notion:

\begin{definition}\label{ConjugateTransformationOfAdjoints}
  **([[conjugate transformations of adjoints]])**
  \linebreak
  Given a [[pair]] of pairs of adjoint functors between the same [[categories]]

  $$
    \array{
    \mathcal{C}
      \underoverset
        {\underset{R_1}{\longleftarrow}}
        {\overset{L_1}{\longrightarrow}}
        {\;\;\; \bot \;\;\;}
    \mathcal{D}
    \\
    \mathcal{C}
      \underoverset
        {\underset{R_2}{\longleftarrow}}
        {\overset{L_2}{\longrightarrow}}
        {\;\;\; \bot \;\;\;}
    \mathcal{D}
    }
  $$

then a [[pair]] of [[natural transformations]] between the adjoints of the same chirality, of this form

$$
  \lambda \,\colon\, L_1 \to L_2
  \;\;\;\;\;\;
  \rho \,\colon\, R_2 \to R_1
$$
is called *conjugate for* [[MacLane (1971), §IV.7 (5)](#MacLane71)] or a *pseudo-transformation of* [[Harpaz & Prasma (2015), Sec. 2.2](#HarpazPrasma15)] the given adjunctions if they make the following [[diagram]] of [[natural transformations]] between [[hom-sets]] [[commuting diagram|commute]]:
$$
  \array{
    \mathcal{C}\big(
      L_2(-)
      ,\,
      (-)
    \big)
    &\overset{\sim}{\longrightarrow}&
    \mathcal{D}\big(
      (-)
      ,\,
      R_2(-)
    \big)
    \\
    \mathllap{{}^{
      \mathcal{C}\big(\lambda_{(-)},\,-\big)
    }}
    \Big\downarrow
    &&
    \Big\downarrow
    \mathrlap{{}^{
      \mathcal{C}\big(-,\,\rho_{(-)}\big)
    }}
    \\
    \mathcal{C}\big(
      L_1(-)
      ,\,
      (-)
    \big)
    &\overset{\sim}{\longrightarrow}&
    \mathcal{D}\big(
      (-)
      ,\,
      R_1(-)
    \big)
    \mathrlap{\,,}
  }
$$
where the horizontal maps are the given hom-isomorphisms (eq:HomIsomorphismForAdjointFunctors).
\end{definition}
This condition is compatible with [[horizontal composition|horizontal]] and [[vertical composition]] of [[natural transformations]] as [[2-morphisms]] in [[Cat]] and hence yields:

\begin{definition}\label{CatAdj}
The ([[very large category|very large]]) [[wide subcategory|wide]] and [[locally full sub-2-category]] [[CatAdj|$Cat_{Adj}$]] of [[Cat]]
\[
  \label{CatAdjInsideCat}
  Cat_{Adj} \longrightarrow Cat
\]
whose

* [[objects]] are categories,

* [[1-morphisms]] are [[left adjoint]] [[functors]]

* [[2-morphisms]] are [[natural transformations]] which are conjugate in the sense of Def. \ref{ConjugateTransformationOfAdjoints}.

\end{definition}

\begin{proposition}
  Under the [[Grothendieck construction]], the [[Grothendieck fibrations]] which arise from [[pseudofunctors]] $\mathcal{B} \longrightarrow Cat$ that factor through $Cat_{Adj}$ (eq:CatAdjInsideCat) are equivalently the [[bifibrations]].
\end{proposition}
This may be [[category theory]] [[folklore]]; a proof has been spelled out in [Harpaz & Prasma (2015), Prop. 2.2.1](#HarpazPrasma15).

## Properties
 {#Properties}

### Basic properties

+-- {: .num_prop #UniquenessOfAdjoints}
###### Proposition
**([[adjoint functors]] are unique up to [[natural isomorphism]])**

The [[left adjoint]] or [[right adjoint]] to a [[functor]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}), if it exists, is unique up to  [[natural isomorphism]].

=--

+-- {: .proof}
###### Proof

Suppose the functor $L \colon \mathcal{D} \to \mathcal{C}$ is given, and we are asking for uniqueness of its right adjoint, if it exists. The other case is directly analogous.

Suppose that $R_1, R_2 \;\colon\; \mathcal{C} \to \mathcal{D}$ are two [[functors]] which are [[right adjoint]] to $L$. Then for each $d \in \mathcal{D}$ the corresponding two hom-isomorphisms (eq:HomIsomorphismForAdjointFunctors) combine to say that there is a [[natural isomorphism]]

$$
  \Phi_d
  \;\colon\;
  Hom_{\mathcal{C}}(-,R_1(d))
  \;\simeq\;
  Hom_{\mathcal{C}}(-,R_2(d))
$$

As in the proof of Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject}, the [[Yoneda lemma]] implies that

$$
  \Phi_d \;=\; y( \phi_d )
$$

for some [[isomorphism]]

$$
  \phi_d \;\colon\;  R_1(d) \overset{\simeq}{\to} R_2(d)
  \,.
$$

But then the uniqueness statement of Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} implies that the collection of these isomorphisms for each object constitues a [[natural isomorphism]] between the functors.

=--

+-- {: .num_prop #AdjointsPreserveCoLimits}
###### Proposition
**([[left adjoints preserve colimits and right adjoints preserve limits]])**

Let $(L \dashv R) \colon \mathcal{D} \to \mathcal{C}$ be a pair of [[adjoint functors]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}). Then

* $L$ [[preserved limit|preserves]] all [[colimits]] that exist in $\mathcal{C}$,

* $R$ preserves all [[limits]] in $\mathcal{D}$.

=--

+-- {: .proof}
###### Proof

Let $y : I \to \mathcal{D}$ be a [[diagram]] whose [[limit]] $\lim_{\leftarrow_i} y_i$ exists. Then we have a sequence of [[natural isomorphism]]s, natural in $x \in C$

$$
  \begin{aligned}
    Hom_{\mathcal{C}}(x, R {\lim_\leftarrow}_i y_i)
    &  \simeq
    Hom_{\mathcal{D}}(L x, {\lim_\leftarrow}_i y_i)
    \\
    & \simeq
    {\lim_\leftarrow}_i Hom_{\mathcal{D}}(L x, y_i)
    \\
    & \simeq
    {\lim_\leftarrow}_i Hom_{\mathcal{C}}( x, R y_i)
    \\
    & \simeq
    Hom_{\mathcal{C}}( x, {\lim_\leftarrow}_i R y_i)
    \,,
  \end{aligned}
$$

where we used the hom-isomorphism (eq:HomIsomorphismForAdjointFunctors) and the fact that any [[hom-functor]] preserves limits (see there). Because this is natural in $x$ the [[Yoneda lemma]] implies that we have an [[isomorphism]]

$$
  R {\lim_\leftarrow}_i y_i
  \simeq
  {\lim_\leftarrow}_i R y_i
  \,.
$$

The argument that shows the preservation of colimits by $L$ is analogous.

=--

+-- {: .num_remark #AdjointFunctorTheorem}
###### Remark

A partial converse to Prop. \ref{AdjointsPreserveCoLimits} is provided by the [[adjoint functor theorem]]. See also _[Pointwise Expression](#PointwiseExpression)_ below.

=--

+-- {: .num_prop #FullyFaithfulAndInvertibleAdjoints}
###### Proposition

Let $L \dashv R$ be a pair of adjoint functors (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}). Then the following holds:

* $R$ is [[faithful functor|faithful]] precisely if the component of the [[unit of an adjunction|counit]] over every object $x$ is an [[epimorphism]] $L R x \stackrel{}{\to} x $;

* $R$ is [[full functor|full]] precisely if the component of the [[unit of an adjunction|counit]] over every object $x$ is a [[split monomorphism]] $L R x \stackrel{}{\to} x $;

* $L$ is [[faithful functor|faithful]] precisely if the component of the [[unit of an adjunction|unit]] over every object $x$ is a [[monomorphism]] $x \hookrightarrow R L x $;

* $L$ is [[full functor|full]] precisely if the component of the [[unit of an adjunction|unit]] over every object $x$ is a [[split epimorphism]] $x \to R L x $;

* $R$ is [[full and faithful functor|full and faithful]]
  (exhibits a [[reflective subcategory]])
  precisely if
  the [[unit of an adjunction|counit]] is a [[natural isomorphism]]
  $\epsilon : L \circ R \stackrel{\simeq}{\to} Id_D$

* $L$ is [[full and faithful functor|full and faithful]]
  (exhibits a [[coreflective subcategory]]) precisely if
  the [[unit of an adjunction|unit]]
  is a natural isomorphism
  $\eta : Id_C \stackrel{\simeq}{\to} R \circ L$.

* The following are equivalent:

  * $L$ and $R$ are both [[full and faithful functor|full and faithful]];

  * $L$ is an [[equivalence of categories|equivalence]];

  * $R$ is an [[equivalence of categories|equivalence]].

=--

{#FurtherProperties} (Further properties are listed in [this MathOverflow discussion](https://mathoverflow.net/questions/100808/properties-of-functors-and-their-adjoints).)

|   |    |    |
|---|----|----|
| $\phantom{A}$**[[adjunction]]**$\phantom{A}$  |  |  $\phantom{A}$[[adjunction unit|unit]] is [[isomorphism|iso]]:$\phantom{A}$  |
|  |  | $\phantom{A}$[[coreflective subcategory|coreflection]]$\phantom{A}$ |
| $\phantom{A}$[[adjunction counit|counit]] is [[isomorphism|iso]]:$\phantom{A}$ | $\phantom{A}$[[reflective subcategory|reflection]]$\phantom{A}$ | $\phantom{A}$[[adjoint equivalence of categories|adjoint equivalence]]$\phantom{A}$  |
{: style='margin:auto'}

+-- {: .proof}
###### Proof

For the characterization of faithful $R$ by epi counit components, notice (as discussed at _[[epimorphism]]_ ) that $L R x \to x$ being an epimorphism is equivalent to the induced [[function]]

$$
  Hom(x, a) \to Hom(L R x, a)
$$

being an [[injection]] for all objects $a$. Then use that, by adjointness, we have an isomorphism

$$
  Hom(L R x , a ) \stackrel{\simeq}{\to} Hom(R x, R a)
$$

and that, by the formula for [[adjuncts]] and the [[zig-zag identity]], this is  such that the composite

$$
 R_{x,a} :  Hom(x,a) \to Hom(L R x, a) \stackrel{\simeq}{\to}
    Hom(R x, R a)
$$

is the component map of the functor $R$ ([this Prop.](geometry+of+physics+–+basic+notions+of+category+theory#ReExpressingMiddleFunctorInAdjointTriple)):

$$
  \begin{aligned}
    (x \stackrel{f}{\to} a) & \mapsto
    (L R x \to x \stackrel{f}{\to} a)
    \\
    & \mapsto
    (R L R x \to R x \stackrel{R f}{\to} R a)
    \\
    & \mapsto
    (R x \to R L R x \to R x \stackrel{R f}{\to} R a)
    \\
    & = (R x \stackrel{R f}{\to} R a)
  \end{aligned}
  \,.
$$

Therefore $R_{x,a}$ is injective for all $x,a$, hence $R$ is faithful, precisely if $L R x \to x$ is an epimorphism for all $x$. The characterization of $R$ full is just the same reasoning applied to the fact that $\epsilon_x \colon L R x \to x$ is a [[split monomorphism]] iff for all objects $a$ the induced function
\[
  Hom(x, a) \to Hom(L R x, a)
\]
is a surjection.

For the characterization of faithful $L$ by monic units notice that analogously (as discussed at [[monomorphism]]) $x \to R L x$ is a monomorphism if for all objects $a$ the function

$$
  Hom(a,x ) \to Hom(a, R L x)
$$

is an injection. Analogously to the previous argument we find that this is equivalent to

$$
  L_{a,x} : Hom(a,x ) \to Hom(a, R L x) \stackrel{\simeq}{\to} Hom(L a, L x)
$$

being an injection. So $L$ is faithful precisely if all $x \to R L x$ are monos. For $L$ full, it's just the same applied to $x \to R L x$ [[split epimorphism]] iff the induced function
$$
  Hom(a,x ) \to Hom(a, R L x)
$$
is a surjection, for all objects $a$.

The proof of the other statements proceeds analogously.

=--

Parts of this statement can be strengthened:

+-- {: .num_prop #ReflectionRecognizedByIdempotency}
###### Proposition

Let $(L \dashv R) : D \to C$ be a pair of [[adjoint functors]] such that there is _any_ [[natural isomorphism]]

$$
  L R \simeq Id
  \,,
$$

then also the [[unit of an adjunction|counit]] $\epsilon : L R \to Id$ is an [[isomorphism]].

=--

This appears as ([Johnstone, lemma 1.1.1](#Johnstone)).

+-- {: .proof}
###### Proof

Using the given [[isomorphism]], we may transfer the [[comonad]] structure on $L R$ to a comonad structure on $Id_D$. By the [[Eckmann-Hilton argument]] the [[endomorphism monoid]] of $Id_D$ is commutative. Therefore,  since the coproduct on the comonad $Id_D$ is a [[left inverse]] to the counit (by the co-[[unitality]] property applied to this degenerate situation), it is in fact a two-sided [[inverse]] and hence the $Id_D$-counit is an [[isomorphism]]. Transferring this back one finds that also the counit of the comand $L R$, hence of the adjunction $(L \dashv R)$ is an isomorphism.

=--

### Pointwise expression
 {#PointwiseExpression}

+-- {: .num_prop #PointwiseExpressionOfLeftAdjoints}
###### Proposition
**(pointwise expression of [[left adjoints]] in terms of [[limits]] over [[comma categories]])**

A [[functor]] $R \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$  has a [[left adjoint]] $L \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ precisely if

1. $R$ [[preserved limit|preserves]] all [[limits]] that exist in $\mathcal{C}$;

1. for each [[object]] $d \in \mathcal{D}$, the [[limit]] of the canonical functor out of the [[comma category]] of $R$ under $d$

   $$
     d/R \longrightarrow \mathcal{C}
   $$

   exists.

In this case the value of the [[left adjoint]] $L$ on $d$ is given by that limit:

\[
  \label{FormulaForLeftAdjointByPointwiseLimit}
  L(d)
  \;\simeq\;
  \underset{\underset{ \left( c,  \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right)  \in d/R }{\longleftarrow}}{\lim} c
\]

=--

(e.g. [[Categories Work|MacLane, chapter X, theorem 2]])

+-- {: .proof}
###### Proof

First assume that the left adjoint exist. Then

1. $R$ is a [[right adjoint]] and hence preserves limits since all [[right adjoints preserve limits]];

1. by Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor} the [[adjunction unit]] provides a [[universal morphism]] $\eta_d$ into $L(d)$, and hence, by Prop. \ref{UniversalMorphismsAreInitialObjectsInCommaCategory}, exhibits $(L(d), \eta_d)$ as the [[initial object]] of  the [[comma category]] $d/R$. The limit over any category with an initial object exists, as it is given by that initial object.

Conversely, assume that the two conditions are satisfied and let $L(d)$ be given by (eq:FormulaForLeftAdjointByPointwiseLimit). We need to show that this yields a left adjoint.

By the assumption that $R$ preserves all limits that exist, we have

\[
  \label{RAppliedtoFormulaForLeftAdjointByPointwiseLimit}
  \array{
    R(L(d))
    & =
    R\left(
    \underset{\underset{ \left( c,  \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right)  \in d/R }{\longleftarrow}}{\lim} c
  \right)
   \\
   & \simeq
    \underset{\underset{ \left( c,  \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right)  \in d/R }{\longleftarrow}}{\lim} R(c)
  }
\]

Since the $d \overset{f}{\to} R(d)$ constitute a [[cone]] over the [[diagram]] of the $R(d)$, there is universal morphism

$$
  d \overset{\phantom{AA} \eta_d \phantom{AA}}{\longrightarrow} R(L(d))
  \,.
$$

By Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor} it is now sufficient to show that $\eta_d$ is a [[universal morphism]] into $L(d)$, hence that for all $c \in \mathcal{C}$ and $d \overset{g}{\longrightarrow} R(c)$ there is a unique morphism $L(d) \overset{\widetilde f}{\longrightarrow} c$ such that

$$
  \array{
    && d
    \\
    & {}^{\mathllap{ \eta_d }}\swarrow && \searrow^{\mathrlap{f}}
    \\
    R(L(d)) && \underset{\phantom{AA}R(\widetilde f)\phantom{AA}}{\longrightarrow} && R(c)
    \\
    L(d)
      &&\underset{\phantom{AA}\widetilde f\phantom{AA}}{\longrightarrow}&&
    c
  }
$$

By Prop. \ref{UniversalMorphismsAreInitialObjectsInCommaCategory}, this is equivalent to $(L(d), \eta_d)$ being the [[initial object]] in the [[comma category]] $c/R$, which in turn is equivalent to it being the [[limit]] of the [[identity functor]] on $c/R$ ([this prop.](initial+object#LimitOverIdentityFunctorIsInitialObject)). But this follows directly from the limit formulas (eq:FormulaForLeftAdjointByPointwiseLimit) and (eq:RAppliedtoFormulaForLeftAdjointByPointwiseLimit).

=--

See at _[[adjoint functor theorem]]_ for more.

[[!include relation between adjunctions and monads -- section]]

### Opposite adjoint functors

Given a pair of [[adjoint functors]]

$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\;\;\;\bot\;\;\;\;}
  \mathcal{C}
$$

there is an induced *[[opposite adjunction]]* of [[opposite functors]]  between their [[opposite categories]] of the form

$$
  \mathcal{D}^{op}
    \underoverset
      {\underset{L^{op}}{\longleftarrow}}
      {\overset{R^{op}}{\longrightarrow}}
      {\;\;\;\;\bot\;\;\;\;}
  \mathcal{C}^{op}
  \,.
$$

Hence where $L$ was the [[left adjoint]], its [[opposite functor|opposite]] becomes the [[right adjoint]], and dually for $R$.

This is immediate from the definition of [[opposite categories]] and the characterization of adjoint functors via the corresponding [hom-isomorphism](#InTermsOfHomIsomorphism).

The [[adjunction unit]] of the opposite adjunction has as components the components of the original [[adjunction counit]], regarded in the opposite category, and dually:

$$
  \epsilon^{R^{op} L^{op}}_{c}
  \;\colon\;
  R^{op}\circ L^{op}(c)
  \xrightarrow{\;\;  \big( \eta^{R L}_c \big) ^{op}  \;\;}
  c
  \,,
  {\phantom{AAAAAA}}
  \eta^{L^{op} R^{op}}_{d}
  \;\colon\;
  d
  \xrightarrow{\;\;  \big( \epsilon^{L R}_d \big) ^{op}  \;\;}
  L^{op} \circ R^{op}(d)
  \,.
$$

### Composing adjunctions

Given two pairs of [[adjoint functors]]

$$
  \mathcal{E}
    \underoverset
      {\underset{R'}{\longrightarrow}}
      {\overset{L'}{\longleftarrow}}
      {\;\;\;\;\bot\;\;\;\;}
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\;\;\;\bot\;\;\;\;}
  \mathcal{C}
$$

there is an induced *composite adjunction* $L' \circ L \dashv R \circ R'$ between $\mathcal{C}$ and $\mathcal{E}$.

This is immediate from the characterization of adjoint functors in terms of [hom-isomorphisms](#InTermsOfHomIsomorphism):

$$
  Hom_{\mathcal{E}}(L'(L(-)),-) \;\simeq\; Hom_{\mathcal{D}}(L(-),R'(-)) \;\simeq\; Hom_{\mathcal{C}}(-,R(R'(-)))
$$

### Pre- and postcomposite adjoint functors
  {#PrePostcompositeAdjunctions}

Given a pair of [[adjoint functors]]

$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\;\;\;\bot\;\;\;\;}
  \mathcal{C}
$$

and a [[category]] $\mathcal{E}$, there is an induced adjunction of precomposition functors between the [[functor categories]] $[\mathcal{C}, \mathcal{E}]$ and $[\mathcal{D}, \mathcal{E}]$ of the form

$$
  [\mathcal{D}, \mathcal{E}]
    \underoverset
      {\underset{- \circ L}{\longrightarrow}}
      {\overset{- \circ R}{\longleftarrow}}
      {\;\;\;\;\bot\;\;\;\;}
  [\mathcal{C}, \mathcal{E}]
  \,.
$$

Hence where $L$ was the [[left adjoint]], its precomposition functor $- \circ L$ becomes the [[right adjoint]], and dually for $R$.

The components $\eta_F : F \Rightarrow F \circ R \circ L$ of the [[unit of an adjunction|unit]] and $\epsilon_F : F \circ L \circ R \Rightarrow F$ of the [[counit of an adjunction|counit]] are given by [[whiskering]] the original unit and counit with $F$ on the left.

By uniqueness of adjoints, this implies that [[Kan extension|left Kan extensions]] along $L$ are given by precomposition with $R$, which is another way of saying that $R$ is the [absolute left Kan extension](#in_terms_of_kan_extensionsliftings) of the [[identity functor]] along $L$. Dually, [[Kan extension|right Kan extensions]] along $R$ are given by precomposition with $L$.

There is also an induced adjunction of postcomposition functors between $[\mathcal{E}, \mathcal{C}]$ and $[\mathcal{E}, \mathcal{D}]$ of the form

$$
  [\mathcal{E}, \mathcal{D}]
    \underoverset
      {\underset{R \circ -}{\longrightarrow}}
      {\overset{L \circ -}{\longleftarrow}}
      {\;\;\;\;\bot\;\;\;\;}
  [\mathcal{E}, \mathcal{C}]
  \,.
$$

The components $\eta_F : F \Rightarrow R \circ L \circ F$ of the [[unit of an adjunction|unit]] and $\epsilon_F : L \circ R \circ F \Rightarrow F$ of the [[counit of an adjunction|counit]] are given by [[whiskering]] the original unit and counit with $F$ on the right.

By uniqueness of adjoints, this implies that [[Kan lift|left Kan lifts]] along $R$ are given by postcomposition with $L$, which is another way of saying that $L$ is the [absolute left Kan lift](#in_terms_of_kan_extensionsliftings) of the [[identity functor]] along $R$. Dually, [[Kan lift|right Kan lifts]] along $L$ are given by postcomposition with $R$.

[[!include sliced adjoint functors -- section]]

## Examples
 {#Examples}

The central point about examples of adjoint functors is:

_Adjoint functors are ubiquitous_ .

To a fair extent, [[category theory]] is all about adjoint functors and the other [[universal construction]]s: [[Kan extension]]s, [[limit]]s, [[representable functor]]s, which are all special cases of adjoint functors -- and adjoint functors are special cases of these.

Listing examples of adjoint functors is much like listing examples of [[integral]]s in [[analysis]]: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see [[coend]] for more).

Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.

### General

* A pair of adjoint functors between [[posets]] is a _[[Galois correspondence]]_.

* A pair of adjoint functors $(L \dashv R)$ where $R$ is a [[full and faithful functor]] exhibits a [[reflective subcategory]].

  In this case $L$ may be regarded as a [[localization]]. The fact that the adjunction provides universal factorization through unit and counit in this case means that every morphism $f : c \to R d$ into a local object factors through the localization of $c$.

* A pair of adjoint functors that is also an [[equivalence of categories]] is called an [[adjoint equivalence]].

* A pair of adjoint functors where $C$ and $D$ have finite [[limit]]s and $L$ preserves these finite limits is a [[geometric morphism]]. These are one kind of morphisms between [[topos]]es. If in addition $R$ is full and faithful, then this is a [[geometric embedding]].

* The left and right adjoint functors $p_!$ and $p_*$ (if they exist) to a functor $p^* : [K',C] \to [K,C]$ between [[functor categories]] obtained by precomposition with a functor $p : K \to K'$ of [[diagram]] categories are called the left and right [[Kan extension]] functors along $p$

  $$
    (Lan_p \dashv p^* \dashv Ran_p)
    :=
    (p_! \dashv p^* \dashv p_*) :
    [K,C]
     \stackrel{\overset{p_!}{\to}}{\stackrel{\overset{p^*}{\leftarrow}}{\underset{p_*}{\to}}}
    [K',C]
    \,.
  $$

  If $K' = {*}$ is the [[terminal category]] then this are the [[limit]] and [[colimit]] functors on $[K,C]$.

  If $C = $ [[Set]] then this is the [[direct image]] and [[inverse image]] operation on [[presheaves]].

* if $R$ is regarded as a [[forgetful functor]] then its left adjoint $L$ is a regarded as a [[free functor]].

* If $C$ is a category with small [[colimit]]s and $K$ is a [[small category]] (a [[diagram]] category)  and $Q : K \to C$ is any functor, then this induces a [[nerve and realization]] pair of adjoint functors

  $$
    (|-|_Q \dashv N_Q) : C \stackrel{\overset{|-|_Q}{\leftarrow}}{\underset{N_Q}{\to}}
    [K^{op}, Set]
  $$

  between $C$ and the [[category of presheaves]] on $K$, where

  * the [[nerve]] functor is given by

    $$
      N_Q(c) := Hom_C(Q(-),c) : k \mapsto Hom_C(Q(k),c)
    $$

  * and the realization functor is given by the [[coend]]

    $$
      |F|_Q := \int^{k \in K} Q(k)\cdot F(k)

      \,,
    $$

    where in the integrand we have the canonical [[copower|tensoring]] of
    $C$ over [[Set]] ($Q(k) \cdot F(k) = \coprod_{s \in F(k)} Q(k)$).

  A famous examples of this is obtained for $C = $ [[Top]], $K = \Delta$
  the [[simplex category]] and $Q : \Delta \to Top$ the functor that sends
  $[n]$ to the standard topological $n$-[[simplex]]. In this case
  the nerve functor is the
  [[fundamental infinity-groupoid|singular simplicial complex]] functor
  and the realization is ordinary [[geometric realization]].

## Related concepts
 {#RelatedConcepts}

* **adjoint functor**, [[adjunction]]

  * [[strong adjoint functor]]

  * {#RelatedSelfAdjointFunctor} [[self-adjoint functor]]

  * [[adjoint triple]], [[adjoint quadruple]], [[recollement]]

  * [[dual adjunction]]

  * [[adjunction of two variables]]

  * [[ambidextrous adjunction]]

  * [[adjoint monad]]

  * [[fixed point of an adjunction]]

  * [[enriched adjunction]]

* [[transformation of adjoints]]

  [[CatAdj|$Cat_{Adj}$]]

* [[proadjoint]], [[Hopf adjunction]]

* [[2-adjunction]]

  [[adjoint 2-functor]], [[strict adjoint 2-functor]]

  [[biadjunction]], [[lax 2-adjunction]], [[pseudoadjunction]]

* [[adjoint (∞,1)-functor]]

* [[(∞,n)-category with adjoints]]

* [[multi-adjoint]], [[weak adjoint]], [[solution set condition]]

## References
 {#References}

For the basics, see any text on [[category theory]] (and see the references at _[[adjunction]]_), for instance:

* {#MacLane71} [[Saunders MacLane]], Chapter IV of: _[[Categories Work|Categories for the working mathematician]]_, Graduate texts in mathematics, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Borceux94} [[Francis Borceux]], Section 3 in Volume 1 and Section 4.2 in Volume 2 of _[[Handbook of Categorical Algebra]]_ (1994)

* {#Johnstone} [[Peter Johnstone]], first pages of _[[Sketches of an Elephant]]_ (2022)

* _[[geometry of physics -- categories and toposes]] -- [Adjunctions](https://ncatlab.org/nlab/show/geometry+of+physics+--+categories+and+toposes#Adjunctions)_

Though the definition of an [[adjoint equivalence]] appears in [[Grothendieck|Grothendieck's]] [[Tohoku]] paper, the idea of adjoint functors in general goes back to

* {#Kan58} [[Daniel Kan]], *Adjoint functors*, Transactions of the American Mathematical Society **87** 2 (1958) 294-329 [[jstor:1993102](http://www.jstor.org/stable/1993102)]

and in [relation](monad#RelationToAdjunctionsAndMonadicity) to [[comonad|co]]/[[monads]] to

* [[Peter J. Huber]], §4 in: *Homotopy theory in general categories*, Mathematische Annalen **144** (1961) 361–385 [[doi:10.1007/BF01396534](https://doi.org/10.1007/BF01396534)]

and its fundamental relevance for [[category theory]] was highlighted in

* {#Freyd64} [[Peter Freyd]], _Abelian categories -- An introduction to the theory of functors_, Harper's  Series  in  Modern  Mathematics, Harper  &  Row,   New  York (1964) [[pdf](http://www.maths.ed.ac.uk/~aar/papers/freydab.pdf)]

* {#Lawvere69} [[William Lawvere]], _Adjointness in Foundations_, ([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296

The history of the idea that adjoint functors formalize aspects of [[dialectics]] is recounted in

* {#Lambek82} [[Joachim Lambek]], _The Influence of Heraclitus on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982) ([doi:10.1007/978-94-009-8462-2_6](https://doi.org/10.1007/978-94-009-8462-2_6))

  (more along these lines at _[[adjoint modality]]_)

See also:

* Wikipedia, [Adjoint Functors](http://en.wikipedia.org/wiki/Adjoint_functors)

* [[The Catsters]] ([list](http://www.youtube.com/view_play_list?p=54B49729E5102248))

More on the notion of [[transformation of adjoints]]:

* {#HarpazPrasma15} [[Yonatan Harpaz]], [[Matan Prasma]], Section 2.2. of: _The Grothendieck construction for model categories_, Advances in Mathematics **281** (2015) 1306-1363 [[arXiv:1404.1852](https://arxiv.org/abs/1404.1852), [10.1016/j.aim.2015.03.031](https://doi.org/10.1016/j.aim.2015.03.031)]

  > (in the context of the [[model structure on a Grothendieck construction]])

[[!redirects adjoint functors]]

[[!redirects adjoint pair of functors]]
[[!redirects adjoint pairs of functors]]

[[!redirects universal arrow]]
[[!redirects universal arrows]]

[[!redirects universal morphism]]
[[!redirects universal morphisms]]


adjoint functor theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

_Adjoint functor theorems_ are theorems stating that under certain conditions a [[functor]] that preserves [[limit]]s is a [[right adjoint]], and that a functor that preserves [[colimit]]s is a [[left adjoint]].

A basic result of [[category theory]] is that right [[adjoint functors]] [[preserved limit|preserve]] all [[limits]] that exist in their [[domain]], and, dually, left adjoints preserve all [[colimits]].  An _adjoint functor theorem_ is a statement that (under certain conditions) the converse holds: a functor which preserves limits is a right adjoint.

The basic idea of an adjoint functor theorem is that _if_ we could assume that a [[large category]] $D$ had all [[limits]] over [[small category|small]] and [[large category|large]] [[diagrams]], then for $R : D \to C$ a [[functor]] that [[preserved limit|preserves]] all these limits we might define its would-be left adjoint $L$ by  taking  $L c$ to be the limit
$$
  L c \coloneqq \lim_{c\to R d} d
$$
This notation stands for a limit over the [[comma category]] $c/R$ (whose [[objects]] are [[pairs]] $(d,f:c\to R d)$ and whose [[morphisms]] are arrows $d\to d'$ in $D$ making the obvious triangle [[commuting diagram|commute]] in $C$) of the projection functor $\pi: c/R \to D$ that forgets the morphism $f$:
$$
  L c = \lim_{c\to R d} \pi
  \,.
$$

Because with this definition there would be, for every $d$, an obvious morphism
$$
  L R d \stackrel{=}{\to} \lim_{R d \to R d'} d' \to d
$$
(the component map over $d$ of the limiting [[cone]]). Moreover, because $R$ preserves limits, we would have an [[isomorphism]]
$$
  R L c \simeq \lim_{c\to R d} R d
$$
(the limit of the functor $R \pi$), and hence an obvious morphism of [[cone]] tips
$$
 c \to R L c
 \,.
$$

It is easy to check that these would be the [[unit of an adjunction|unit]] and [[counit of an adjunction|counit]] of an [[adjunction]] $L\dashv R$. See _[[adjoint functor]]_ for more.

The problem with this would-be argument is that in general the comma category $(c/G)$ may not be [[small category]]. But one can generally not expect a large category to have all large limits: even if we pass to a [[universe]] in which $(c/G)$ is considered small, a [classical theorem of Freyd](complete+small+category#CompleteSmallCategoriesArePosets) says that any [[complete small category]] is a [[preorder]] (see [[complete small category]] for the proof, which is valid in [[classical logic]] and also holds classically in any [[Grothendieck topos]]).  Thus, the argument we gave above is necessarily only an **adjoint functor theorem for preorders**:

+-- {: .num_theorem #ForPosets}
###### Theorem

If $G:D\to C$ is any functor between (small) preorders such that $D$ has, and $G$ preserves, all small [[meets]], then $G$ has a left adjoint.

=--

(This theorem holds in [[constructive mathematics]], although not in [[predicative mathematics]]; the classical reasoning before this explains why the theorem is not more general, but the proof itself is already constructive.)

To obtain adjoint functor theorems for categories that are not preorders, one must therefore impose various additional "size conditions" on the category $D$ and/or the functor $G$.

## Statement

+-- {: .num_theorem #StandardAdjointFunctorTheorem}
###### Theorem

Sufficient conditions for a limit-preserving functor $R : C \to D$ to be a [[right adjoint]] include:

* $C$ is [[complete category|complete]] and [[locally small category|locally small]], and $R$ satisfies the [[solution set condition]].

  This is Freyd's original version, sometimes called the "**General Adjoint Functor Theorem**".

* $C$ is complete, locally small, [[well-powered category|well-powered]], and has a [[small set|small]] [[cogenerating set]], and $D$ is [[locally small category|locally small]].

  This is sometimes called the "**Special Adjoint Functor Theorem**", and abbreviated to SAFT.

* $C$ is locally small and [[total category|cototal]], and $D$ is locally small.

=--

In the first two cases, which work by replacing large limits by small ones, it suffices to assume that $R$ preserves small limits (that it preserves all limits will follow).  The third case works by assuming that $C$ has, while not all large limits, enough so that the theorem goes through; thus is this case $R$ must be already known to preserve [[large category|large]] limits as well.

+-- {: .proof}
###### Proof

Here is a proof of the General Adjoint Functor Theorem: that a functor $R : C \to D$ out of a [[locally small category]] $C$ with all small limits has a left adjoint if it preserves these [[limit]]s and satisfies the [[solution set condition]].

From the discussion at adjoint functors -- In terms of universal arrows we have that the existence of the adjoint is equivalent to the existence for each $d \in D$ of an [[initial object]] $i_d : d \to R L d$ in the [[comma category]] $(d \downarrow R)$: an object such that for each $f : d \to R d'$ there is a unique $\tilde f$ such that

$$
  \array{
    && d
    \\
    & {}^{\mathllap{i_d}}\swarrow && \searrow^{\mathrlap{f}}
    \\
    R L d &&\underset{R \tilde f}{\to}&& R d'
    \\
    \\
    L d &&\underset{\tilde f}{\to}&& d'
  }
$$

commutes. Now an initial object is the limit of the identity functor, but this is generally a large limit; we replace this with some small limit conditions that guarantee existence of an initial object.

1. Let $Y$ be a category. Call a small family of objects $F$ _weakly initial_ if for every object $y$ of $Y$ there exists $x \in F$ and a morphism $f: x \to y$.

1. Suppose $Y$ has small products. If $F$ is a [[weak multilimit|weakly initial family]], then $\prod_{x \in F} x$ is a [[weakly initial object]].

1. **Claim:** Suppose $Y$ is locally small and has joint equalizers of small families. If $x$ is a weakly initial object, then the domain $e$ of the joint equalizer $i: e \to x$ of all arrows $x \to x$ is an initial object. **Proof:** clearly $e$ is weakly initial. Suppose given an object $y$ and arrows $f, g: e \to y$; we must show $f = g$. Let $j: d \to e$ be the equalizer of $f$ and $g$. There exists an arrow $k: x \to d$. The arrow $i: e \to x$ equalizes $1_x$ and $i j k: x \to x$, so $i j k i = i$. Since $i$ is monic, $j (k i) = 1_e$. Thus $j$ is an epi, and $f = g$ follows.

If $C$ is locally small and small-complete and $R: C \to D$ preserves limits, then $d \downarrow R$ is locally small and small-complete for every object $d$ of $D$.

If in addition each $d \downarrow R$ has a weakly initial family (solution set condition), then by 2. and 3. each $d \downarrow R$ has an initial object. This restates the condition that $R$ has a left adjoint.

=--

In fact, it suffices for $C$ to be [[Cauchy complete]] and for $R$ to be $\alpha$-[[flat functor|flat]] for every $\alpha$ (i.e. for its [[Yoneda extension]] to be [[continuous functor|continuous]]). See [Borceux](#Bouceux).

+-- {: .proof}
###### Proof of SAFT
As before, the proof proceeds by constructing initial objects of comma categories. We assume that $C$ is locally small, small-complete, well-powered, has a cogenerating set $\{c_\alpha: \alpha \in A\}$, and that $R: C \to D$ is a small-[[continuous functor]] into a [[locally small category]] $D$.

As before, for each object $d$ of $D$, the comma category $d \downarrow R$ is locally small and small-complete. Moreover, it is easy to check that it is well-powered, and that the set of all objects of the form $d \to R c_\alpha$ is a cogenerating set for $d \downarrow R$.

It then remains to prove that any locally small, small-complete, well-powered category $X$ with a cogenerating set $\{k_s: s \in S\}$ has an initial object. The initial object $0$ is constructed as the intersection = pullback of all subobjects of $\prod_s k_s$, i.e., the minimal subobject. Then, given $f, g: 0 \to x$, the equalizer $Eq(f, g)$ is isomorphic to $0$ because $0$ is minimal, and so $f = g$: there is at most one arrow $0 \to x$ for each $x$.

On the other hand, for each $x$ the canonical map

$$i: x \to \prod_{s \in S} k_{s}^{\hom(x, k_s)}$$

is monic since the $k_s$ cogenerate. The following pullback of $i$,

$$\array{
k & \to & x \\
\downarrow & & \downarrow \mathrlap{i} \\
\prod_s k_{s}^1 & \stackrel{\prod_s k_{s}^!}{\to} & \prod_s k_{s}^{\hom(x, k_s)}
},$$
gives a subobject $k$ of $\prod_s k_s$ that maps to $x$, and into which $0$ embeds. Thus there exists a map $0 \to x$, and we conclude $0$ is initial.
=--

### In locally presentable categories

In practice an important special case is that of functors between
[[locally presentable categories]]. For these there is the following version of an adjoint functor theorem.

+-- {: .num_theorem #AdjFuncTheoremForLocallyPresentableCats}
###### Theorem

Let $F \colon C \to D$ be a functor between [[locally presentable categories]]. Then

* $F$ has a [[right adjoint]] if and only if it preserves all small [[colimits]].

* $F$ has a [[left adjoint]] if and only if it is an [[accessible functor]] and preserves all small [[limits]].

=--

The second statement, characterizing when $F$ has a left adjoint, is  ([AdamekRosicky, theorem 1.66](#AdamekRosicky)). In the "if" direction, this is an application of the general adjoint functor theorem: any accessible functor satisfies the solution set condition. The "only if", particularly that having a left adjoint forces accessibility, takes a little work. But in any case there are easy examples that show that continuity alone is insufficient, i.e., examples of [[continuous functors]] between [[locally presentable categories]] that do not have [[left adjoints]]. See below in the section [In locally presentable categories](#InLocallyPresentableCategories).

The first statement, characterizing when $F$ has a right adjoint, can be proven using the special adjoint functor theorem: by a non-trivial theorem ([AdamekRosicky, theorem 1.58](#AdamekRosicky)), any locally presentable category is co-wellpowered.

Thus, the first statement can be strengthened by removing the assumption that $D$ is locally presentable: it is enough that $D$ be locally small. For example, if $C$ is locally presentable, then every [[continuous functor]]

$$C^{op} \to Set$$

has a left adjoint (is representable), because its opposite $C \to Set^{op}$ is cocontinuous and therefore has a right adjoint, even though $Set^{op}$ is not locally presentable.

A [[right adjoint]] to any [[cocontinuous functor]] $F \colon C \to D$ between [[locally presentable categories]] can also be constructed directly. If $C$ is locally $\lambda$-presentable and $P_\lambda$ is the subcategory of $\lambda$-small objects, then $C$ is equivalent to the full subcategory of $[P_\lambda^{op},Set]$ of presheaves that preserve $\lambda$-small limits ([AdamekRosicky, theorem 1.46](#AdamekRosicky)). The presheaves in the image of the functor $D \to [P_\lambda^{op},Set]$ defined by $d \mapsto \hom(F-,d)$ preserve $\lambda$-small limits because $F$ is cocontinuous. So this functor factors through the subcategory $C$. The functor $D \to C$ so-constructed is a right adjoint to $F$.

## Examples {#Examples}

### In locally presentable categories
 {#InLocallyPresentableCategories}

The following is a counter-example, indicating the need for something more than just continuity to force a functor between locally presentable categories to be a right adjoint; as stated in theorem \ref{AdjFuncTheoremForLocallyPresentableCats}, the missing extra condition is precisely accessibility.

+-- {: .num_example}
###### Example

For every infinite [[cardinal number]] $\kappa$, let
$G_\kappa$ be a [[simple group]] of [[cardinality]] $\kappa$.
Define the functor $ML:$ [[Group]] $\to$ [[Set]] to be the
[[product]] of all the [[representable functors]]
$Hom(G_\kappa,-)$.
Since no group can admit a nontrivial
[[homomorphism]] from proper-[[class]]-many of the $G_\kappa$,
this functor does indeed land (or can be redefined to land) in [[Set]].
Since it is a product of representables, it is
[[continuous functor|continuous]]
(and of course [[Group]] and [[Set]] are
[[locally presentable categories]]),
but it is not itself representable (hence has no [[left adjoint]]).

=--

[[André Joyal]] has been attributing this example to [[Saunders MacLane]]; it appears in print for instance right at the beginning of ([AdámekKoubekTrnková01](#AdamekKoubekTrnkova)).

### In cocomplete categories

Suppose $C$ and $D$ are categories that admit [[small colimits]]
(i.e., are [[cocomplete]])
and $F\colon C\to D$ is a functor that preserves [[small colimits]]
(i.e., is cocontinuous).
The $F$ has a [[right adjoint functor|right adjoint]]
if and only if for any object $d\in D$
the functor
$$C^{op} \to Set$$
that sends
$$c\mapsto D(F(c),d)$$
is a [[small presheaf]] on $C$.

See the [MathOverflow answer](https://mathoverflow.net/questions/346153/surmounting-set-theoretical-difficulties-in-algebraic-geometry/346162#346162) by [[Ivan Di Liberti]].

### In toposes

+-- {: .num_prop}
###### Proposition

Every [[sheaf topos]] is a [[total category]] and a [[cototal category]].

=--

See the discussion at _[[Grothendieck topos]]_.

It follows that

+-- {: .num_cor}
###### Corollary

Let $F : C \to D$ be a [[functor]] between [[sheaf toposes]]. Then

* $F$ has a [[right adjoint]] precisely if it preserves all small [[colimits]];

* $F$ has a [[left adjoint]] precisely if it preserves all small [[limits]].

=--

### In presheaf categories
 {#InPresheafCategories}

It is instructive to spell out the construction of the [[right adjoint]] from a colimit preserving functor $L$ in the simple case where all categories are [[categories of presheaves]]. This is a particularly simple case, but is useful in itself and serves as a template for the general case.

So let now $C$ and $D$ be [[small categories]] and $L$ a [[colimit]]-[[preserved colimit|preserving]] functor between their [[categories of presheaves]] (which we abbreviate $\widehat C \;\coloneqq\; [C^{op}, Set]$, etc):

$$
  L \;\colon\; \widehat{C} \longrightarrow \widehat{D}
$$

Then its [[right adjoint]] $ R \;\colon\; \widehat{D} \longrightarrow \widehat{C}$ is given (with $y_c \coloneqq C(-, c)$ denoting the [[Yoneda embedding]]) by

\[
  \label{RightAdjointOfColimitPreservingFunctorOfPresheaves}
  R
    \;\colon\;
  A
    \;\mapsto\;
  R(A)
   \;\coloneqq\;
   \widehat{D}
   \left(
     L(y_{(-)}),
     A
   \right)
   \,,
\]

as we shall check in a moment. But first notice that using the [[co-Yoneda lemma]] this may be rewritten as

$$
  \cdots
    \;\simeq\;
   \int^{c \in C}
   \widehat{D}
   \left(
     L(y_c), A
   \right)
     \cdot y_c
   \,,
$$

where the [[coend]] is equivalently given by the [[colimit]]

$$
  \cdots \;=\; \lim_{\underset{L c \to A}{\to}} c
  \,.
$$

This is the formula for the would-be right adjoint from the general discussion above, only that here the colimit is only over the representables, hence over a small category.

Now we check that the functor $R$ thus obtained is indeed right adjoint to $L$, by explicitly checking the hom-isomorphism ([here](adjoint+functor#InTermsOfHomIsomorphism)) of the pair of [[adjoint functors]]:

We compute $\widehat{D}(L(X),A)$. In the first step

$$
  \widehat{D}(L(X), A)
  \simeq
  \widehat{D}
  \left(
     L
     \left(
        \int^{c \in C} X(c) \cdot y_c
     \right),
     A
   \right)
$$

we use the [[co-Yoneda lemma]] for $X$. Then, because $L$ [[preserved limit|preserves]] colimits, this is

$$
  \cdots
    \simeq
   \widehat{D}
   \left(
     \int^c X(c) \cdot L(y_c), A
   \right)
  \,.
$$

Since the [[hom-functor]] preserves limits in both arguments, we can take the [[coend]] out to get an [[end]]

$$
  \cdots
    \simeq
   \int_{c \in C}
     \widehat{D}(X(c) \cdot L(y_c), A)
  \,.
$$

Then we use the standard [[copower|tensoring]] of our categories over [[Set]] to get

$$
  \cdots
    \simeq
   \int_{c \in C}
   Set\left(
     X(c),
     \widehat{D}(L(y_c),A)
   \right)
  \,.
$$

And finally this is recognized as the formula for the [[hom-set]] of presheaves (see the discussion at _[[functor category]]_):

$$
  \cdots
    \simeq
   \widehat{C}
   \left(
     X,
     \widehat{D}(L(y_{(-)}),A)
    \right)
     \;=\;
    \widehat{C}(X, R A)
  \,,
$$

where on the right we identified (eq:RightAdjointOfColimitPreservingFunctorOfPresheaves).

The [[composition]] of this sequence of [[natural isomorphisms]] is hence the desired hom-isomorphism:

$$
  \widehat{D}(L(X), A) \simeq \widehat{C}(X, R(A))
  \,.
$$

## Related concepts

* ([[cocontinuous functor|co]])[[continuous functor]]

* [[indexed adjoint functor theorem]]

* [[adjoint (∞,1)-functor theorem]]

* [[adjoint triangle theorem]]

* [[adjoint lifting theorem]]

## References

The classical adjoint functor theorems originate in the exercise section of ch.3 (pp.84ff) in

* [[Peter Freyd]], *Abelian Categories*, Harper & Row New York 1964. (Reprinted with author's comment as [TAC reprints no. 3 (2003)](http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html))

A more recent exposition is in

* [[Peter Freyd]], Andre Scedrov, *Categories, Allegories*, North-Holland, 1990. (Also Dover reprint New York 2014, pp.144-148)

where the [[solution set condition]] is called "pre-adjointness".

Careful discussions can be found in

* {#Bouceux} [[Francis Borceux]], *[[Handbook of Categorical Algebra]] I*, Cambridge UP, 1994. (sections 3.3, 6.6)

* [[Saunders MacLane]], §V.6, V.8 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

A brief introductory discussion is around [theorem 5.4](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf#page=54) of

* [[Jaap van Oosten]], *Basic category theory*, ms. ([pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf))

A detailed expository survey is

* Oliver Kullmann, *The adjoint functor theorem*, ms. ([pdf slides](http://www.cs.swan.ac.uk/~csoliver/Hauptseminar/Quellen/200702Swansea_b.pdf))

The adjoint functor theorem in context with [[Yoneda embedding]] is discussed in

* Friedrich Ulmer, *The adjoint functor theorem and the Yoneda embedding*, Illinois J. Math. **15**  no.3 (1971), pp. 355-361. ([euclid](http://projecteuclid.org/euclid.ijm/1256052605))

The connection between the solution set condition and the Čech homology construction is discussed in

* Renato Betti, _Čech methods and the adjoint functor theorem_ , Cah. Top. Géom. Diff. Cat. **XXVI** no.3 (1985) pp.245-257. ([numdam](http://www.numdam.org/item/?id=CTGDC_1985__26_3_245_0))

An enriched adjoint functor theorem is given in:

* [[Francis Borceux]]. *Limites enrichies et existence de $ V $-foncteur adjoint*. Cahiers de topologie et géométrie différentielle catégoriques 16.4 (1975): 395-408. ([pdf](http://www.numdam.org/item/CTGDC_1975__16_4_395_0.pdf))

The parallels between the adjoint functor theorem for categories and the computation of colimits from limits in a lattice as well as similar parallels between co/completeness for Boolean algebras and Paré's theorem for finite completeness of toposes is studied from a type-theoretical perspective in

* [[Dusko Pavlovic|Duško Pavlović]], _On completeness and cocompleteness in and around small categories_ , APAL **74** (1995) pp.121-152.

The case for [[locally presentable categories]] is discussed in

* [[Jiří Adámek]], [[Jiri Rosicky]], *[[Locally presentable and accessible categories]]*, Cambridge UP, 1994.
  {#AdamekRosicky}

* [[Jiří Adámek]], V. Koubek and V. Trnková, *How large are left exact functors?*, Theory and Applications of Categories **8** (2001), pp. 377-390. ([abstract](http://www.emis.de/journals/TAC/volumes/8/n13/8-13abs.html))
 {#AdamekKoubekTrnkova}

A specialization to the locally $\alpha$-presentable case is given in Theorem 2.11 of

* [[Giacomo Tendas]], *On continuity of accessible functors*, Applied Categorical Structures, 2022, [doi](https://doi.org/10.1007/s10485-022-09677-x).

A relative version of Freyd's classical results is in

* [[Brian Day]], *An adjoint-functor theorem over topoi*, Bull. Austral. Math. Soc. **15** (1976), pp. 381-394.

[[indexed adjoint functor theorem|Adjoint functor theorems for indexed categories]] are discussed in

* [[Robert Paré]], [[Dietmar Schumacher]], _Abstract Families and the Adjoint Functor Theorems_ , pp.1-125 in LNM **661** Springer Heidelberg 1978.

A history of the general adjoint functor theorem is discussed in:

* [[Hans-E. Porst]], _The history of the General Adjoint Functor Theorem_ [[arxiv:2310.19528](https://arxiv.org/abs/2310.19528)]

A stronger version for finitary functors between locally finitely presentable categories whose domain is ranked, requiring only the preservation of countable limits for the existence of a left adjoint, is discussed in

* [[Jiří Adámek]], Lurdes Sousa, _A Finitary Adjoint Functor Theorem_, [arXiv](https://arxiv.org/abs/2311.14965).

[[!redirects adjoint functor theorems]]


adjoint lifting theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Statement ##

+-- {: .un_theorem}
###### Theorem
**(The adjoint lifting theorem)**. Consider the following [[commutative diagram|commutative square]] of [[functor]]s:
$$
\begin{array}{cccc}\mathcal{A} & \overset{Q}{\to}  & \mathcal{B}    \\
^{U}\downarrow     &                   & \downarrow^{V} \\
\mathcal{C}        & \underset{R}{\to} & \mathcal{D}
\end{array}
$$

and suppose that

* $U$ and $V$ are [[monadic functor|monadic]], and

* $\mathcal{A}$ has [[coequalizer]]s of reflexive pairs.

Then, if $R$ has a [[left adjoint]], then $Q$ also has a [[left adjoint]].
=--

A detailed proof may be found in Sec. 4.5 of Vol. 2 of [Borceux](#Borceux)
(see especially Theorem 4.5.6 on p. 226 and Ex. 4.8.6 on p. 252). Also ([Johnstone, prop. 1.1.3](#Johnstone)) For a sketch of proof, see ahead.

+-- {: .un_cor}
###### Corollary
If the bottom functor $R$ of the above square is the
identity arrow (so that $U=V\circ Q$), if $U$ and $V$ are monadic, and
if $\mathcal{A}$ has coequalizers of reflexive pairs, then $Q$ is monadic.
=--

+-- {: .proof}
###### Proof
The adjoint lifting theorem implies the existence of a left
adjoint, and the rest is a straightforward application of the
[[monadicity theorem]].
=--

## Sketch of proof ##

We may assume the situation of the following diagram (with $V Q = R U$):
$$
\begin{array}{cccc}\mathcal{C}^\mathbb{T} & \underoverset{Q}{K(?)}{\leftrightarrows}  & \mathcal{D}^\mathbb{S}    \\
^{U}\downarrow \uparrow^{F}    &                   & ^{G}\uparrow\downarrow^{V} \\
\mathcal{C}        & \underoverset{R}{L}{\leftrightarrows}  & \mathcal{D}
\end{array}
$$
where
$\mathbb{T}=\langle T,\varepsilon\colon 1_{\mathcal{C}}\Rightarrow
T,\mu \rangle$ is a monad on
$\mathcal{C}$, $\mathbb{S}=\langle S,\zeta\colon 1_{\mathcal{D}}\Rightarrow
S,\eta\rangle$ is a monad on
$\mathcal{D}$, $U$ and $V$ are the forgetful functors, and $F$ and $G$
are the free algebra functors.

Let us write $\tau\colon F U\Rightarrow 1_{\mathcal{C}^{\mathbb{T}}}$ for
the counit of the adjunction $F\dashv U$ and $\sigma\colon
G V\Rightarrow 1_{\mathcal{D}^{\mathbb{S}}}$ for the counit of the
adjunction $G\dashv V$. As usual, we have $T = U F$, $S = V G$,
$\mu=U\tau F$, and $\eta=V\sigma G$.

Finally, let $L$ be a left adjoint to $R$ (which
exists by assumption), and let $\alpha\colon
1_{\mathcal{D}}\Rightarrow R L$ and $\beta\colon LR\Rightarrow
1_{\mathcal{C}}$ be the unit and counit (respectively) of the
adjunction $L\dashv R$.

We would like to construct a functor $K\colon
\mathcal{D}^{\mathbb{S}}\to \mathcal{C}^{\mathbb{T}}$. To get a hint
of what $K$ should look like, let us assume for a moment that such a
$K$ already exists.  In this case, we have $K G\dashv V Q(=R U)$. But
$F L$ is also left adjoint to $R U$, and by the uniqueness of a left
adjoint we must have $K G = F L$ (at least up to a natural isomorphism).

From this, we already know how to define $K$ on free algebras. Also,
being a left adjoint, $K$ preserves in particular all coequalizers. But
every $\mathbb{S}$-algebra $\langle D,\xi\colon SD\to D\rangle$ is
the (object part) of a reflexive coequalizer, namely, the [[canonical
presentation]]
$$
G S(D)\underoverset{G\xi}{\sigma_{G D}}{\rightrightarrows}G(D)\overset{\sigma_{\langle
D,\xi\rangle}=\xi}{\rightarrow}\langle D,\xi \rangle
$$
Applying $K$ and using $K G=F L$, we see that $K\langle D,\xi\rangle$
should be the object part of a reflexive coequalizer in
$\mathcal{C}^{\mathbb{T}}$ of the form
$$
F L S(D)\underoverset{F L\xi}{?}{\rightrightarrows}F L(D)\overset{x}{\rightarrow}K\langle
D,\xi \rangle
$$
(recall that we assume that $\mathcal{C}^{\mathbb{T}}$ has
coequalizers of reflexive pairs).

To eventually define $K\langle D,\xi \rangle$ as a coequalizer (as above),
we first need some reasonable guess for the ∞-arrow. For this, we will
need a lemma.

+-- {: .un_lemma}
###### Lemma
There exists a natural transformation $\lambda\colon S R
\Rightarrow R T$ for which the following diagram of functors and
natural transformations is commutative:

$$
\begin{array}{ccccccc} R & \overset{\zeta R}{\to} & S R &
 &  \overset{\eta R}{\leftarrow}& & S S R \\
 & ^{R\varepsilon}\searrow & ^{\lambda}\downarrow & & & & ^{S\lambda}\downarrow\\
 & & R T & \overset{R\mu}{\leftarrow} & R T T & \overset{\lambda T}{\leftarrow} & S R T
\end{array}
$$
=--

+-- {: .proof}
###### Proof
Define $\lambda := V\sigma Q F \circ V G R\varepsilon$, so that
$$
\lambda \colon
S R = V G R\overset{V G R\varepsilon}{\to}V G R T = V G R U F = V G V Q
F\overset{V\sigma Q F}{\to} V Q F = R U F = R T.
$$
The required commutativity may be verified by using the commutative
diagrams in the definitions of a monad and an EM-algebra,
naturality, and the triangular identities. For details, see the proof
of Lemma 4.5.1 of [Borceux](#Borceux),
pp. 222-223. (Note that this lemma does not depend on the
existence of a left adjoint for the bottom horizontal arrow, nor on
the existence of coequalizers. Only the commutativity is required.)
=--

We may now return to our task of defining the ∞-arrow in the diagram
preceding the lemma. We would like to get from $F L S(D)$ to $F L(D)$,
and for this, we will construct a natural transformation
$F L S\Rightarrow F L$ in the following way.  First, we have
$$
S\overset{S\alpha}{\to} S R L \overset{\lambda L}{\to} R T L.
$$
Applying $L$ and composing with $\beta T L$, we get
$$
LS\overset{L\lambda L\circ LS\alpha}{\longrightarrow}L R T L \overset{\beta TL}{\to} TL.
$$
Applying $F$ and composing with $\tau F L$, we finally get
$$
F L S\overset{F\beta T L\circ F L\lambda L\circ F L
S\alpha}{\longrightarrow}F T L = F U F L\overset{\tau F L}{\to} F L
$$
Let us call the resulting natural transformation $\omega$, that is,
$$
\omega:=\tau F L \circ F\beta T L\circ F L\lambda L\circ F L S\alpha.
$$

Now we take the sought for ∞-arrow to be $\omega_D$, and
*define* $K\langle D,\xi\rangle$ as the object of some fixed
coequalizer of $\omega_D$ and $F L\xi$:
$$
F L S(D)\underoverset{F L\xi}{\omega_D}{\rightrightarrows}F
L(D)\overset{x}{\rightarrow}K\langle D,\xi \rangle.
$$
In order to do this, we must first verify that the parallel
arrows above have a common section (since we only assume that
$\mathcal{C}^{\mathbb{T}}$ has coequalizers of reflexive pairs). To
find a guess for a common section, note that the common section
for the parallel pair in the above canonical presentation in
$\mathcal{D}^{\mathbb{S}}$ is $G\zeta_{V\langle
D,\xi\rangle}=G\zeta_D$, and if $K$ exists, then applying $K$ gives
$K G\zeta_D=F L\zeta_D$. Having this guess, it is now straightforward
to verify that $F L\zeta_D$ is indeed a common section, as required.

So, we have defined an object function of a would be left adjoint $K$.
To make it into a functor left adjoint to $Q$, we will build a
universal arrow from $\langle D,\xi\rangle$ to $Q$, whose object part
is $K\langle D,\xi\rangle$ (Theorem IV.1.2(ii) of [[Categories
Work]]).

To get an arrow $\langle D,\xi\rangle\to Q K\langle D,\xi\rangle$ ,
suppose for a moment that we have a natural
transformation $\varphi\colon G\Rightarrow Q F L$ such that  $\varphi\circ
\sigma G = Q\omega\circ \varphi S$.  Then the left square in the following
diagram commutes:
$$
\begin{array}{ccccc}G S(D)&
\underoverset{G\xi}{\sigma_{G D}}{\rightrightarrows} & G(D) &\overset{\sigma_{\langle
D,\xi\rangle}=\xi}{\rightarrow}&\langle D,\xi \rangle \\
^{\varphi_{S D}}\downarrow && ^{\varphi_D}\downarrow && ^{\chi}\downarrow\\
Q F L S D &\underoverset{Q F L\xi}{Q\omega_D}{\rightrightarrows} & Q F
L D &\overset{Q x}{\rightarrow}&Q K\langle D,\xi\rangle
\end{array}
$$
Since both rows are forks, it follows that $Q x\circ \varphi_D$ has
the same composition with the arrows of the upper
parallel pair, and hence there exists a unique arrow $\chi\colon
\langle D,\xi\rangle\to Q K\langle D,\xi\rangle$ making the right
square commutative (recall that the upper row is a coequalizer).

It is now possible to prove that the pair $\langle K\langle
D,\xi\rangle,\chi \rangle$ is a universal arrow from $\langle
D,\xi\rangle$ to $Q$, showing that $K$ is indeed (the object
function) of a left adjoint (for details, see the proof of Theorem
4.5.6, pp. 226-227 in [Borceux](#Borceux)).

But we still have to prove the existence of a natural
transformation $\varphi\colon G\Rightarrow Q F L$ such that  $\varphi\circ
\sigma G = Q\omega\circ \varphi S$. For this, we define
$\varphi:=\sigma Q F L \circ G R\varepsilon L \circ G\alpha$. Since
$V$ is faithful, to prove the required property of $\varphi$, it is
enough to prove that $V\varphi \circ V\sigma G = V Q\omega\circ
V\varphi S$, and this is a long, yet straightforward,
computation (noting that $V\varphi = \lambda L \circ VG\alpha$ and using
the commutative diagram from Lemma 1; see Lemma 4.5.3, p. 224 of [Borceux](#Borceux)).

## Examples ##

### Forgetful functors between varieties of algebras ###
Since varieties of algebras are [[cocompleteness of varieties of algebras|cocomplete]] and monadic over
$\mathbf{Set}$, the corollary implies that forgetful functors between
varieties of algebras (e.g., the forgetful functor
$\mathbf{Rng}\to\mathbf{Ab}$) are monadic.

### Sufficient conditions for cocompleteness of monadic categories ###
Let $\mathcal{J}$ be an arbitrary category, and consider the commutative diagram
$$
\begin{array}{ccccc}\mathcal{A} & \overset{\Delta}{\to} & \mathcal{A}^\mathcal{J}    \\
^{U}\downarrow                 &                        & \downarrow^{U^{\mathcal{J}}} \\
\mathcal{C}                    & \underset{\Delta}{\to} & \mathcal{C}^\mathcal{J}
\end{array}
$$
where $U$ is monadic, $\Delta$ is the [[diagonal functor]] and $U^{\mathcal{J}}=U\circ
-$. If $F$ is left adjoint
to $U$, then $F^\mathcal{J}$ is left adjoint to $U^{\mathcal{J}}$ (using
the unit and counit of the original adjunction, one can construct
appropriate natural transformations that satisfy the triangular
identities, see, e.g., p. 119 of [[Categories Work]]).  Also, the
conditions of the monadicity theorem for $U$ imply those for
$U^\mathcal{J}$ (basically because the definition of a split fork involves only
compositions and identities, and because natural transformations are composed
componentwise).

Now, if $\mathcal{C}$ is $\mathcal{J}$-cocomplete (so that the
bottom horizontal functor has a left adjoint) and $\mathcal{A}$ has
coequalizers of reflexive pairs, then the adjoint lifting theorem
implies that $\mathcal{A}$ is $\mathcal{J}$-cocomplete. In particular,
if $\mathcal{A}$ has coequalizers of reflexive pairs and $\mathcal{C}$
is small-cocomplete, then $\mathcal{A}$ is small cocomplete.

## Related pages

* The adjoint lifting theorem is a corollary of the [[adjoint triangle theorem]].

* [[adjoint functor theorem]]

## References ##

* [[Michael Barr]], [[Charles Wells]], section 3.7, pp.131 in: *[[Toposes, Triples, and Theories]]*, Springer (1985), Reprints in Theories and Applications of Categories **12** (2005) 1-287 [[tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)]

* [[Francis Borceux]], _[[Handbook of Categorical Algebra]] II_ , Cambridge UP 1994. (section 4.5, pp.221ff)
{#Borceux}

* {#Johnstone} [[Peter Johnstone]], *Adjoint lifting theorems for categories of algebras*, Bull. London Math. Soc. **7** (1975) 294-297 [[doi:10.1112/blms/7.3.294](https://doi.org/10.1112/blms/7.3.294)]

* [[Peter Johnstone]], section A1.1, p.5 in: *[[Sketches of an Elephant]] I*, Oxford UP 2002

On the dual theorem for [[comonads]]:

* William F. Keigher, _Adjunctions and comonads in differential algebra_, Pacific J. Math. **59,** 1 (1975) 99-112 [[euclid:pjm/1102905501](http://projecteuclid.org/euclid.pjm/1102905501)]

* [[John Power]], _A unified approach to the lifting of adjoints_, [[Cahiers]] **XXIX**  1 (1988)  67-77 [[numdam](http://www.numdam.org/item/CTGDC_1988__29_1_67_0)]

[[!redirects adjoint lifting theorems]]


adjoint logic

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Adjoint logic_ or _adjoint type theory_ is [[formal logic]] or [[type theory]] which natively expresses [[adjunctions]] of [[modal operators]], [[adjoint modalities]].

## Related pages

* [temporal logic](temporal+logic#temporal_logic_in_terms_of_adjoints)

## References

* [[Nick Benton]], [[Philip Wadler]], _Linear logic, monads and the lambda calculus_, In IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press, 1996.

* Jason Reed, _A judgemental deconstruction of modal logic_, 2009, ([pdf](http://www.cs.cmu.edu/~jcreed/papers/jdml2.pdf))

* Klaas Pruiksma, William Chargin, [[Frank Pfenning]], and Jason Reed, *Adjoint Logic*, 2018, [[pdf](https://www.cs.cmu.edu/~fp/papers/adjoint18b.pdf), [[PCPR18-AdjointLogic.pdf:file]]]

A framework for ([[homotopy type theory|homotopy]]-)[[type theory|type theoretic]] adjoint logic ([[modal type theory]]) is discussed, in various stages of generality, in

* {#Shulman15} [[Mike Shulman]], _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

  (specifically for [[cohesive homotopy type theory]])

* {#LicataShulman} [[Dan Licata]], [[Mike Shulman]], _Adjoint logic with a 2-category of modes_, in _[Logical Foundations of Computer Science 2016](http://lfcs.info/lfcs-2016/)_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf), [slides](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint-lfcs-slides.pdf))

  (for [[modal type theory|modal]] [[unary type theory]])

* [[Daniel Licata]], [[Mike Shulman]], and [[Mitchell Riley]], _A Fibrational Framework for Substructural and Modal Logics (extended version)_, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) ([doi: 10.4230/LIPIcs.FSCD.2017.25](http://drops.dagstuhl.de/opus/volltexte/2017/7740/), [pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf))

  (for [[modal type theory|modal]] [[simple type theory]])

Review includes

* {#Licata18} [[Dan Licata]], _Synthetic Mathematics in Modal Dependent Type Theories_, tutorial at _[Types, Homotopy Theory and Verification](https://www.him.uni-bonn.de/programs/current-trimester-program/types-sets-constructions/workshop-types-homotopy-type-theory-and-verification/)_, 2018 ([pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/him-tutorial.pdf))

[[!redirects adjoint logics]]

[[!redirects adjoint type theory]]
[[!redirects adjoint type theories]]