hcju/stacksqa huggingface dataset:
For $i = 1, 2$ an $i$-morphism of stacks is the same thing as a $i$-morphism of fibred categories, see Definition \ref{definition-stacks-over-C}. By Lemma \ref{lemma-adjointness-pullback-pushforward} we have already $$ \Mor_{\textit{Fib}/\mathcal{C}}(\mathcal{S}, u^p\mathcal{T}) = \Mor_{\textit{Fib}/\mathcal{D}}(u_p\mathcal{S}, \mathcal{T}) $$ Hence the result follows from Lemma \ref{lemma-stackify-universal-property-more} as $u^p\mathcal{T} = f_*\mathcal{T}$ and $f^{-1}\mathcal{S}$ is the stackification of $u_p\mathcal{S}$.
It is clear that $u$ is a continuous functor. Hence we obtain functors $f_{Zar, *} = u^s$ and $f_{Zar}^{-1} = u_s$, see Sites, Section \ref{sites-section-morphism-sites}. To see that we obtain a morphism of sites we have to show that $u_s$ is exact. We will use Sites, Lemma \ref{sites-lemma-directed-morphism} to see this. Let $V \subset Y_n$ be an open subset. The category $\mathcal{I}_V^u$ (see Sites, Section \ref{sites-section-functoriality-PSh}) consists of pairs $(U, \varphi)$ where $\varphi : [m] \to [n]$ and $U \subset X_m$ open such that $Y(\varphi)(V) \subset f_m^{-1}(U)$. Moreover, a morphism $(U, \varphi) \to (U', \varphi')$ is given by a $\psi : [m'] \to [m]$ such that $X(\psi)(U) \subset U'$ and $\varphi \circ \psi = \varphi'$. It is our task to show that $\mathcal{I}_V^u$ is cofiltered. \medskip\noindent We verify the conditions of Categories, Definition \ref{categories-definition-codirected}. Condition (1) holds because $(X_n, \text{id}_{[n]})$ is an object. Let $(U, \varphi)$ be an object. The condition $Y(\varphi)(V) \subset f_m^{-1}(U)$ is equivalent to $V \subset f_n^{-1}(X(\varphi)^{-1}(U))$. Hence we obtain a morphism $(X(\varphi)^{-1}(U), \text{id}_{[n]}) \to (U, \varphi)$ given by setting $\psi = \varphi$. Moreover, given a pair of objects of the form $(U, \text{id}_{[n]})$ and $(U', \text{id}_{[n]})$ we see there exists an object, namely $(U \cap U', \text{id}_{[n]})$, which maps to both of them. Thus condition (2) holds. To verify condition (3) suppose given two morphisms $a, a': (U, \varphi) \to (U', \varphi')$ given by $\psi, \psi' : [m'] \to [m]$. Then precomposing with the morphism $(X(\varphi)^{-1}(U), \text{id}_{[n]}) \to (U, \varphi)$ given by $\varphi$ equalizes $a, a'$ because $\varphi \circ \psi = \varphi' = \varphi \circ \psi'$. This finishes the proof.
The first part is immediate from the definitions. For the second part, note that in the proof of Lemma \ref{lemma-simplicial-space-site-functorial} we have shown that for a $V \subset Y_n$ open the category $(\mathcal{I}_V^u)^{opp}$ contains as a cofinal subcategory the category of opens $U \subset X_n$ with $f_n^{-1}(U) \supset V$ and morphisms given by inclusions. Hence we see that the restriction of $u_p\mathcal{F}$ to opens of $Y_n$ is the presheaf $f_{n, p}\mathcal{F}_n$ as defined in Sheaves, Lemma \ref{sheaves-lemma-pullback-presheaves}. Since $f_{Zar}^{-1}\mathcal{F} = u_s\mathcal{F}$ is the sheafification of $u_p\mathcal{F}$ and since sheafification uses only coverings and since coverings in $Y_{Zar}$ use only inclusions between opens on the same $Y_n$, the result follows from the fact that $f_n^{-1}\mathcal{F}_n$ is (correspondingly) the sheafification of $f_{n, p}\mathcal{F}_n$, see Sheaves, Section \ref{sheaves-section-presheaves-functorial}.
Besides the properties of our functor mentioned in the statement, the category $X_{n, Zar}$ has fibre products and equalizers and the functor commutes with them (beware that $X_{Zar}$ does not have all fibre products). Hence the lemma follows from the discussion in Sites, Sections \ref{sites-section-cocontinuous-functors} and \ref{sites-section-cocontinuous-morphism-topoi} and Modules on Sites, Section \ref{sites-modules-section-exactness-lower-shriek}. More precisely, Sites, Lemmas \ref{sites-lemma-cocontinuous-morphism-topoi}, \ref{sites-lemma-when-shriek}, and \ref{sites-lemma-preserve-equalizers} and Modules on Sites, Lemmas \ref{sites-modules-lemma-g-shriek-adjoint} and \ref{sites-modules-lemma-exactness-lower-shriek}.
Part (2) holds for augmentations of simplicial objects in any category. Thus $Y(\varphi)^{-1} a_m^{-1} \mathcal{F} = a_n^{-1}\mathcal{F}$ which defines an $Y(\varphi)$-map from $a_m^{-1}\mathcal{F}$ to $a_n^{-1}\mathcal{F}$. Thus we can use (1) as the definition of $a^{-1}\mathcal{F}$ (using Lemma \ref{lemma-describe-sheaves-simplicial-site}) and (4) as the definition of $a_*$. If this defines a morphism of topoi then part (3) follows because we'll have $g_n^{-1} \circ a^{-1} = a_n^{-1}$ by construction. To check $a$ is a morphism of topoi we have to show that $a^{-1}$ is left adjoint to $a_*$ and we have to show that $a^{-1}$ is exact. The last fact is immediate from the exactness of the functors $a_n^{-1}$. \medskip\noindent Let $\mathcal{F}$ be an object of $\Sh(X)$ and let $\mathcal{G}$ be an object of $\Sh(Y_{Zar})$. Given $\beta : a^{-1}\mathcal{F} \to \mathcal{G}$ we can look at the components $\beta_n : a_n^{-1}\mathcal{F} \to \mathcal{G}_n$. These maps are adjoint to maps $\beta_n : \mathcal{F} \to a_{n, *}\mathcal{G}_n$. Compatibility with the simplicial structure shows that $\beta_0$ maps into $a_*\mathcal{G}$. Conversely, suppose given a map $\alpha : \mathcal{F} \to a_*\mathcal{G}$. For any $n$ choose a $\varphi : [0] \to [n]$. Then we can look at the composition $$ \mathcal{F} \xrightarrow{\alpha} a_*\mathcal{G} \to a_{0, *}\mathcal{G}_0 \xrightarrow{\mathcal{G}(\varphi)} a_{n, *}\mathcal{G}_n $$ These are adjoint to maps $a_n^{-1}\mathcal{F} \to \mathcal{G}_n$ which define a morphism of sheaves $a^{-1}\mathcal{F} \to \mathcal{G}$. We omit the proof that the constructions given above define mutually inverse bijections $$ \Mor_{\Sh(Y_{Zar})}(a^{-1}\mathcal{F}, \mathcal{G}) = \Mor_{\Sh(X)}(\mathcal{F}, a_*\mathcal{G}) $$ This finishes the proof. An interesting observation is here that this morphism of topoi does not correspond to any obvious geometric functor between the sites defining the topoi.
Let $U \subset X_m$ be an object of $X_{Zar}$. Then the value of the complex above on $U$ is the complex of abelian groups $$ \ldots \to \mathbf{Z}[\Mor_\Delta([2], [m])] \to \mathbf{Z}[\Mor_\Delta([1], [m])] \to \mathbf{Z}[\Mor_\Delta([0], [m])] $$ In other words, this is the complex associated to the free abelian group on the simplicial set $\Delta[m]$, see Simplicial, Example \ref{simplicial-example-simplex-simplicial-set}. Since $\Delta[m]$ is homotopy equivalent to $\Delta[0]$, see Simplicial, Example \ref{simplicial-example-simplex-contractible}, and since ``taking free abelian groups'' is a functor, we see that the complex above is homotopy equivalent to the free abelian group on $\Delta[0]$ (Simplicial, Remark \ref{simplicial-remark-homotopy-better} and Lemma \ref{simplicial-lemma-homotopy-equivalence-s-N}). This complex is acyclic in positive degrees and equal to $\mathbf{Z}$ in degree $0$.
Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution. Consider the double complex with terms $$ A^{p, q} = \mathcal{I}^q(X_p) $$ and first differential given by the alternating sum along the maps $d^{p + 1}_i$-maps $\mathcal{I}_p^q \to \mathcal{I}_{p + 1}^q$, see Lemma \ref{lemma-describe-sheaves-simplicial-site}. Note that $$ A^{p, q} = \Gamma(X_p, \mathcal{I}_p^q) = \Mor_{\textit{PSh}}(h_{X_p}, \mathcal{I}^q) = \Mor_{\textit{PAb}}(\mathbf{Z}_{X_p}, \mathcal{I}^q) $$ Hence it follows from Lemma \ref{lemma-simplicial-resolution-Z} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf} that the rows of the double complex are exact in positive degrees and evaluate to $\Gamma(X_{Zar}, \mathcal{I}^q)$ in degree $0$. On the other hand, since restriction is exact (Lemma \ref{lemma-restriction-to-components}) the map $$ \mathcal{F}_p \to \mathcal{I}_p^\bullet $$ is a resolution. The sheaves $\mathcal{I}_p^q$ are injective abelian sheaves on $X_p$ (Lemma \ref{lemma-restriction-injective-to-component}). Hence the cohomology of the columns computes the groups $H^q(X_p, \mathcal{F}_p)$. We conclude by applying Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and \ref{homology-lemma-double-complex-gives-resolution}.
This is the analogue of Cohomology, Lemma \ref{cohomology-lemma-describe-higher-direct-images} or of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images} and we strongly encourge the reader to skip the proof. Choosing an injective resolution of $\mathcal{F}$ on $X_{Zar}$ and using the definitions we see that it suffices to show: (1) the restriction of an injective abelian sheaf on $X_{Zar}$ to $(X \times_Y V)_{Zar}$ is an injective abelian sheaf and (2) $a_*\mathcal{F}$ is equal to the rule $$ V \longmapsto H^0((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}}) $$ Part (2) follows from the following facts \begin{enumerate} \item[(2a)] $a_*\mathcal{F}$ is the equalizer of the two maps $a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1$ by Lemma \ref{lemma-augmentation}, \item[(2b)] $a_{0, *}\mathcal{F}_0(V) = H^0(a_0^{-1}(V), \mathcal{F}_0)$ and $a_{1, *}\mathcal{F}_1(V) = H^0(a_1^{-1}(V), \mathcal{F}_1)$, \item[(2c)] $X_0 \times_Y V = a_0^{-1}(V)$ and $X_1 \times_Y V = a_1^{-1}(V)$, \item[(2d)] $H^0((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}})$ is the equalizer of the two maps $H^0(X_0 \times_Y V, \mathcal{F}_0) \to H^0(X_1 \times_Y V, \mathcal{F}_1)$ for example by Lemma \ref{lemma-simplicial-sheaf-cohomology}. \end{enumerate} Part (1) follows after one defines an exact left adjoint $j_! : \textit{Ab}((X \times_Y V)_{Zar}) \to \textit{Ab}(X_{Zar})$ (extension by zero) to restriction $\textit{Ab}(X_{Zar}) \to \textit{Ab}((X \times_Y V)_{Zar})$ and using Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}.
The statements about the functor are straightforward to verify. The existence of $g$ and $g_!$ follow from Sites, Lemmas \ref{sites-lemma-cocontinuous-morphism-topoi} and \ref{sites-lemma-when-shriek}. The description of $g^{-1}$ is immediate from Sites, Lemma \ref{sites-lemma-when-shriek}. The description of $g_*$ and $g_!$ follows as the functors given are right and left adjoint to $g^{-1}$.
Composition of $(\varphi, f) : U \to V$ with $(\psi, g) : V \to W$ is given by $(\varphi \circ \psi, g \circ u_\psi(f))$. This uses that $u_\psi \circ u_\varphi = u_{\varphi \circ \psi}$. \medskip\noindent Let $\{(\text{id}, f_i) :  U_i \to U\}$ be a covering as in (3) and let $(\varphi, g) : W \to U$ be a morphism with $W \in \Ob(\mathcal{C}_m)$. We claim that $$ W \times_{(\varphi, g), U, (\text{id}, f_i)} U_i = W \times_{g, U, f_i} U_i $$ in the category $\mathcal{C}_{total}$ where the right hand side is the object of $\mathcal{C}_m$ defined in Sites, Remark \ref{sites-remark-cartesian-cocontinuous} which exists by property $P$. Compatibility of this type of fibre product with compositions of functors implies the claim (details omitted). Since the family $\{W \times_{g, U, f_i} U_i \to W\}$ is a covering of $\mathcal{C}_m$ by property $P$ we see that the collection of coverings is stable under base change. The other axioms of a site are immediate.
Case A. Let $u_n : \mathcal{D} \to \mathcal{C}_n$ be the common value of the functors $u_\varphi \circ u_0$ for $\varphi : [0] \to [n]$. Then $u_n$ corresponds to a morphism of sites $a_n : \mathcal{C}_n \to \mathcal{D}$, see Sites, Lemma \ref{sites-lemma-composition-morphisms-sites}. The same lemma shows that for all $\varphi : [m] \to [n]$ we have $a_m \circ f_\varphi = a_n$. \medskip\noindent Case B. Let $u_n : \mathcal{C}_n \to \mathcal{D}$ be the common value of the functors $u_0 \circ u_\varphi$ for $\varphi : [0] \to [n]$. Then $u_n$ is cocontinuous and hence defines a morphism of topoi $a_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{D)}$, see Sites, Lemma \ref{sites-lemma-composition-cocontinuous}. The same lemma shows that for all $\varphi : [m] \to [n]$ we have $a_m \circ f_\varphi = a_n$. \medskip\noindent Consider the functor $a^{-1} : \Sh(\mathcal{D}) \to \Sh(\mathcal{C}_{total})$ which to a sheaf of sets $\mathcal{G}$ associates the sheaf $\mathcal{F} = a^{-1}\mathcal{G}$ whose components are $a_n^{-1}\mathcal{G}$ and whose transition maps $\mathcal{F}(\varphi)$ are the identifications $$ f_\varphi^{-1}\mathcal{F}_m = f_\varphi^{-1} a_m^{-1}\mathcal{G} = a_n^{-1}\mathcal{G} = \mathcal{F}_n $$ for $\varphi : [m] \to [n]$, see the description of $\Sh(\mathcal{C}_{total})$ in Lemma \ref{lemma-describe-sheaves-simplicial-site-site}. Since the functors $a_n^{-1}$ are exact, $a^{-1}$ is an exact functor. Finally, for $a_* : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{D})$ we take the functor which to a sheaf $\mathcal{F}$ on $\Sh(\mathcal{D})$ associates $$ \xymatrix{ a_*\mathcal{F} \ar@{=}[r] & \text{Equalizer}(a_{0, *}\mathcal{F}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, *}\mathcal{F}_1) } $$ Here the two maps come from the two maps $\varphi : [0] \to [1]$ via $$ a_{0, *}\mathcal{F}_0 \to a_{0, *}f_{\varphi, *} f_\varphi^{-1}\mathcal{F}_0 \xrightarrow{\mathcal{F}(\varphi)} a_{0, *}f_{\varphi, *} \mathcal{F}_0 = a_{1, *}\mathcal{F}_1 $$ where the first arrow comes from $1 \to f_{\varphi, *} f_\varphi^{-1}$. Let $\mathcal{G}_\bullet$ denote the constant simplicial sheaf with value $\mathcal{G}$ and let $a_{\bullet, *}\mathcal{F}$ denote the simplicial sheaf having $a_{n, *}\mathcal{F}_n$ in degree $n$. By the usual adjuntion for the morphisms of topoi $a_n$ we see that a map $a^{-1}\mathcal{G} \to \mathcal{F}$ is the same thing as a map $$ \mathcal{G}_\bullet \longrightarrow a_{\bullet, *}\mathcal{F} $$ of simplicial sheaves. By Simplicial, Lemma \ref{simplicial-lemma-augmentation-howto} this is the same thing as a map $\mathcal{G} \to a_*\mathcal{F}$. Thus $a^{-1}$ and $a_*$ are adjoint functors and we obtain our morphism of topoi $a$\footnote{In case B the morphism $a$ corresponds to the cocontinuous functor $\mathcal{C}_{total} \to \mathcal{D}$ sending $U$ in $\mathcal{C}_n$ to $u_n(U)$.}. The equalities $a \circ g_n = f_n$ follow immediately from the definitions.
Case A. Say $h_n$ corresponds to the continuous functor $v_n : \mathcal{C}'_n \to \mathcal{C}_n$. Then we can define a functor $v_{total} : \mathcal{C}'_{total} \to \mathcal{C}_{total}$ by using $v_n$ in degree $n$. This is clearly a continuous functor (see definition of coverings in Lemma \ref{lemma-simplicial-site-site}). Let $h_{total}^{-1} = v_{total, s} : \Sh(\mathcal{C}'_{total}) \to \Sh(\mathcal{C}_{total})$ and $h_{total, *} = v_{total}^s = v_{total}^p : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}'_{total})$ be the adjoint pair of functors constructed and studied in Sites, Sections \ref{sites-section-continuous-functors} and \ref{sites-section-morphism-sites}. To see that $h_{total}$ is a morphism of topoi we still have to verify that $h_{total}^{-1}$ is exact. We first observe that $(g'_n)^{-1} \circ h_{total, *} = h_{n, *} \circ g_n^{-1}$; this is immediate by computing sections over an object $U$ of $\mathcal{C}'_n$. Thus, if we think of a sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$ as a system $(\mathcal{F}_n, \mathcal{F}(\varphi))$ as in Lemma \ref{lemma-describe-sheaves-simplicial-site-site}, then $h_{total, *}\mathcal{F}$ corresponds to the system $(h_{n, *}\mathcal{F}_n, h_{n, *}\mathcal{F}(\varphi))$. Clearly, the functor $(\mathcal{F}'_n, \mathcal{F}'(\varphi)) \to (h_n^{-1}\mathcal{F}'_n, h_n^{-1}\mathcal{F}'(\varphi))$ is its left adjoint. By uniqueness of adjoints, we conclude that $h_{total}^{-1}$ is given by this rule on systems. In particular, $h_{total}^{-1}$ is exact (by the description of sheaves on $\mathcal{C}_{total}$ given in the lemma and the exactness of the functors $h_n^{-1}$) and we have our morphism of topoi. Finally, we obtain $g_n^{-1} \circ h_{total}^{-1} = h_n^{-1} \circ (g'_n)^{-1}$ as well, which proves that the displayed diagram of the lemma commutes. \medskip\noindent Case B. Here we have a functor $v_{total} : \mathcal{C}_{total} \to \mathcal{C}'_{total}$ by using $v_n$ in degree $n$. This is clearly a cocontinuous functor (see definition of coverings in Lemma \ref{lemma-simplicial-cocontinuous-site}). Let $h_{total}$ be the morphism of topoi associated to $v_{total}$. The commutativity of the displayed diagram of the lemma follows immediately from Sites, Lemma \ref{sites-lemma-composition-cocontinuous}. Taking left adjoints the final equality of the lemma becomes $$ h_{total}^{-1} \circ (g'_n)^{Sh}_! = g^{Sh}_{n!} \circ h_n^{-1} $$ This follows immediately from the explicit description of the functors $(g'_n)^{Sh}_!$ and $g^{Sh}_{n!}$ in Lemma \ref{lemma-restriction-to-components-site}, the fact that $h_n^{-1} \circ (f'_\varphi)^{-1} = f_\varphi^{-1} \circ h_m^{-1}$ for $\varphi : [m] \to [n]$, and the fact that we already know $h_{total}^{-1}$ commutes with restrictions to the degree $n$ parts of the simplicial sites.
Let $\mathcal{I}^\bullet$ be a K-injective complex on $\mathcal{C}_{total}$ representing $K$. Then $g_n^{-1}K$ is represented by $g_n^{-1}\mathcal{I}^\bullet = \mathcal{I}_n^\bullet$ which is K-injective by Lemma \ref{lemma-restriction-injective-to-component-site}. We have $(g'_n)^{-1}h_{total, *}\mathcal{I}^\bullet = h_{n, *}g_n^{-1}\mathcal{I}_n^\bullet$ by Lemma \ref{lemma-morphism-simplicial-sites} which gives the desired equality.
The morphism $h$ is defined in Lemma \ref{lemma-morphism-simplicial-sites}. The morphisms $a$ and $a'$ are defined in Lemma \ref{lemma-augmentation-site}. Thus the only thing is to prove the commutativity of the diagram. To do this, we prove that $a^{-1} \circ h_{-1}^{-1} = h_{total}^{-1} \circ (a')^{-1}$. By the commutative diagrams of Lemma \ref{lemma-morphism-simplicial-sites} and the description of $\Sh(\mathcal{C}_{total})$ and $\Sh(\mathcal{C}'_{total})$ in terms of components in Lemma \ref{lemma-describe-sheaves-simplicial-site-site}, it suffices to show that $$ \xymatrix{ \Sh(\mathcal{C}_n) \ar[d]_{a_n} \ar[r]_{h_n} & \Sh(\mathcal{C}'_n) \ar[d]^{a'_n} \\ \Sh(\mathcal{D}) \ar[r]^{h_{-1}} & \Sh(\mathcal{D}') } $$ commutes for all $n$. This follows from the case for $n = 0$ (which is an assumption in Remark \ref{remark-morphism-augmentation-simplicial-sites}) and for $n > 0$ we pick $\varphi : [0] \to [n]$ and then the required commutativity follows from the case $n = 0$ and the relations $a_n = a_0 \circ f_\varphi$ and $a'_n = a'_0 \circ f'_\varphi$ as well as the commutation relations $f'_\varphi \circ h_n = h_0 \circ f_\varphi$.
By Lemma \ref{lemma-restriction-to-components-site} we have $g_n^{-1}\mathcal{O} = \mathcal{O}_n$ and hence we obtain our morphism of ringed topoi. By Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules} we obtain the adjoint $g_{n!}$. To prove the formula for $g_{n!}$ we first define a sheaf of $\mathcal{O}$-modules $\mathcal{H}$ on $\mathcal{C}_{total}$ with degree $m$ component the $\mathcal{O}_m$-module $$ \mathcal{H}_m = \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{G} $$ Given a map $\psi : [m] \to [m']$ the map $\mathcal{H}(\psi) : f_\psi^{-1}\mathcal{H}_m \to \mathcal{H}_{m'}$ is given on components by $$ f_\psi^{-1} f_\varphi^*\mathcal{G} \to f_\psi^* f_\varphi^*\mathcal{G} \to f_{\psi \circ \varphi}^*\mathcal{G} $$ Since this map $f_\psi^{-1}\mathcal{H}_m \to \mathcal{H}_{m'}$ is $\mathcal{O}(\psi) : f_\psi^{-1}\mathcal{O}_m \to \mathcal{O}_{m'}$-semi-linear, this indeed does define an $\mathcal{O}$-module (use Lemma \ref{lemma-describe-sheaves-simplicial-site-site}). Then one proves directly that $$ \Mor_{\mathcal{O}_n}(\mathcal{G}, \mathcal{F}_n) = \Mor_{\mathcal{O}}(\mathcal{H}, \mathcal{F}) $$ proceeding as in the proof of Lemma \ref{lemma-restriction-to-components-site}. Thus $\mathcal{H} = g_{n!}\mathcal{G}$ as desired.
Recall that $g_{n!}\mathcal{G}$ is the $\mathcal{O}$-module whose degree $m$ part is the $\mathcal{O}_m$-module $$ \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{G} $$ Here the morphism of ringed topoi $f_\varphi : (\Sh(\mathcal{C}_m), \mathcal{O}_m) \to (\Sh(\mathcal{C}_n), \mathcal{O}_n)$ uses the map $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ of the statement of the lemma. If these maps are flat, then $f_\varphi^*$ is exact (Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact}). By definition of the site $\mathcal{C}_{total}$ we see that these functors have the desired exactness properties and we conclude.
Follows from Lemma \ref{lemma-morphism-simplicial-sites} and \ref{lemma-restriction-module-to-components-site} by keeping track of the sheaves of rings. A small point is that in order to define $h_n$ as a morphism of ringed topoi we set $h_n^\sharp = g_n^{-1}h^\sharp : g_n^{-1}h_{total}^{-1}\mathcal{O}' \to g_n^{-1}\mathcal{O}$ which makes sense because $g_n^{-1}h_{total}^{-1}\mathcal{O}' = h_n^{-1}(g'_n)^{-1}\mathcal{O}' = h_n^{-1}\mathcal{O}'_n$ and $g_n^{-1}\mathcal{O} = \mathcal{O}_n$. Note that $g_n^*\mathcal{F} = g_n^{-1}\mathcal{F}$ for a sheaf of $\mathcal{O}$-modules $\mathcal{F}$ and similarly for $g'_n$ and this helps explain why $(g'_n)^* \circ h_{total, *} = h_{n, *} \circ g_n^*$ follows from the corresponding statement of Lemma \ref{lemma-morphism-simplicial-sites}.
Recall that $g_n^* = g_n^{-1}$ because $g_n^{-1}\mathcal{O} = \mathcal{O}_n$ by the construction in Lemma \ref{lemma-restriction-module-to-components-site}. In particular $g_n^*$ is exact and $Lg_n^*$ is given by applying $g_n^*$ to any representative complex of modules. Similarly for $g'_n$. There is a canonical base change map $(g'_n)^*Rh_{total, *}K \to Rh_{n, *}g_n^*K$, see Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}. By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-modules-abelian-unbounded} the image of this in $D(\mathcal{C}'_n)$ is the map $(g'_n)^{-1}Rh_{total, *}K_{ab} \to Rh_{n, *}g_n^{-1}K_{ab}$ where $K_{ab}$ is the image of $K$ in $D(\mathcal{C}_{total})$. This we proved to be an isomorphism in Lemma \ref{lemma-direct-image-morphism-simplicial-sites} and the result follows.
We will use the description of the functors $g_{n!}$ in Lemma \ref{lemma-restriction-to-components-site} without further mention. As maps of the complex we take $\sum (-1)^i d^n_i$ where $d^n_i : g_{n!}\mathbf{Z} \to g_{n - 1!}\mathbf{Z}$ is the adjoint to the map $\mathbf{Z} \to \bigoplus_{[n - 1] \to [n]} \mathbf{Z} = g_n^{-1}g_{n - 1!}\mathbf{Z}$ corresponding to the factor labeled with $\delta^n_i : [n - 1] \to [n]$. Then $g_m^{-1}$ applied to the complex gives the complex $$ \ldots \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} \mathbf{Z} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} \mathbf{Z} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} \mathbf{Z} $$ on $\mathcal{C}_m$. In other words, this is the complex associated to the free abelian sheaf on the simplicial set $\Delta[m]$, see Simplicial, Example \ref{simplicial-example-simplex-simplicial-set}. Since $\Delta[m]$ is homotopy equivalent to $\Delta[0]$, see Simplicial, Example \ref{simplicial-example-simplex-contractible}, and since ``taking free abelian sheaf on'' is a functor, we see that the complex above is homotopy equivalent to the free abelian sheaf on $\Delta[0]$ (Simplicial, Remark \ref{simplicial-remark-homotopy-better} and Lemma \ref{simplicial-lemma-homotopy-equivalence-s-N}). This complex is acyclic in positive degrees and equal to $\mathbf{Z}$ in degree $0$.
Observe that $\Hom(\mathbf{Z}, \mathcal{F}) = \Gamma(\mathcal{C}_{total}, \mathcal{F})$ and $\Hom(g_{n!}\mathbf{Z}, \mathcal{F}) = \Gamma(\mathcal{C}_n, \mathcal{F}_n)$. Hence this lemma is an immediate consequence of Lemma \ref{lemma-simplicial-resolution-Z-site} and the fact that $\Hom(-, \mathcal{F})$ is exact if $\mathcal{F}$ is injective.