hcju/stacksqa huggingface dataset:
Let $\mathcal{F}$ be a $\mathcal{H}$-torsor. Consider the free abelian sheaf $\mathbf{Z}[\mathcal{F}]$ on $\mathcal{F}$. It is the sheafification of the rule which associates to $U \in \Ob(\mathcal{C})$ the collection of finite formal sums $\sum n_i[s_i]$ with $n_i \in \mathbf{Z}$ and $s_i \in \mathcal{F}(U)$. There is a natural map $$ \sigma : \mathbf{Z}[\mathcal{F}] \longrightarrow \underline{\mathbf{Z}} $$ which to a local section $\sum n_i[s_i]$ associates $\sum n_i$. The kernel of $\sigma$ is generated by sections of the form $[s] - [s']$. There is a canonical map $a : \Ker(\sigma) \to \mathcal{H}$ which maps $[s] - [s'] \mapsto h$ where $h$ is the local section of $\mathcal{H}$ such that $h \cdot s = s'$. Consider the pushout diagram $$ \xymatrix{ 0 \ar[r] & \Ker(\sigma) \ar[r] \ar[d]^a & \mathbf{Z}[\mathcal{F}] \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{H} \ar[r] & \mathcal{E} \ar[r] & \underline{\mathbf{Z}} \ar[r] & 0 } $$ Here $\mathcal{E}$ is the extension obtained by pushout. From the long exact cohomology sequence associated to the lower short exact sequence we obtain an element $\xi = \xi_\mathcal{F} \in H^1(\mathcal{C}, \mathcal{H})$ by applying the boundary operator to $1 \in H^0(\mathcal{C}, \underline{\mathbf{Z}})$. \medskip\noindent Conversely, given $\xi \in H^1(\mathcal{C}, \mathcal{H})$ we can associate to $\xi$ a torsor as follows. Choose an embedding $\mathcal{H} \to \mathcal{I}$ of $\mathcal{H}$ into an injective abelian sheaf $\mathcal{I}$. We set $\mathcal{Q} = \mathcal{I}/\mathcal{H}$ so that we have a short exact sequence $$ \xymatrix{ 0 \ar[r] & \mathcal{H} \ar[r] & \mathcal{I} \ar[r] & \mathcal{Q} \ar[r] & 0 } $$ The element $\xi$ is the image of a global section $q \in H^0(\mathcal{C}, \mathcal{Q})$ because $H^1(\mathcal{C}, \mathcal{I}) = 0$ (see Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}). Let $\mathcal{F} \subset \mathcal{I}$ be the subsheaf (of sets) of sections that map to $q$ in the sheaf $\mathcal{Q}$. It is easy to verify that $\mathcal{F}$ is a $\mathcal{H}$-torsor. \medskip\noindent We omit the verification that the two constructions given above are mutually inverse.
Let $\underline{\mathbf{Z}}$ denote the constant sheaf $\mathbf{Z}$. As $\textit{Ab}(\mathcal{C}) = \textit{Mod}(\underline{\mathbf{Z}})$ we may apply Lemma \ref{lemma-h1-extensions} twice, and it follows that we have to show $$ \Ext^1_{\textit{Mod}(\mathcal{O})}(\mathcal{O}, \mathcal{F}) = \Ext^1_{\textit{Mod}(\underline{\mathbf{Z}})}( \underline{\mathbf{Z}}, \mathcal{F}_{ab}). $$ Suppose that $0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O} \to 0$ is an extension in $\textit{Mod}(\mathcal{O})$. Then we can use the obvious map of abelian sheaves $1 : \underline{\mathbf{Z}} \to \mathcal{O}$ and pullback to obtain an extension $\mathcal{E}_{ab}$, like so: $$ \xymatrix{ 0 \ar[r] & \mathcal{F}_{ab} \ar[r] \ar@{=}[d] & \mathcal{E}_{ab} \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[r] \ar[d]^{1} & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{E} \ar[r] & \mathcal{O} \ar[r] & 0 } $$ The converse is a little more fun. Suppose that $0 \to \mathcal{F}_{ab} \to \mathcal{E}_{ab} \to \underline{\mathbf{Z}} \to 0$ is an extension in $\textit{Mod}(\underline{\mathbf{Z}})$. Since $\underline{\mathbf{Z}}$ is a flat $\underline{\mathbf{Z}}$-module we see that the sequence $$ 0 \to \mathcal{F}_{ab} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \to \mathcal{E}_{ab} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \to \underline{\mathbf{Z}} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \to 0 $$ is exact, see Modules on Sites, Lemma \ref{sites-modules-lemma-flat-tor-zero}. Of course $\underline{\mathbf{Z}} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} = \mathcal{O}$. Hence we can form the pushout via the ($\mathcal{O}$-linear) multiplication map $\mu : \mathcal{F} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \to \mathcal{F}$ to get an extension of $\mathcal{O}$ by $\mathcal{F}$, like this $$ \xymatrix{ 0 \ar[r] & \mathcal{F}_{ab} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \ar[r] \ar[d]^\mu & \mathcal{E}_{ab} \otimes_{\underline{\mathbf{Z}}} \mathcal{O} \ar[r] \ar[d] & \mathcal{O} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{E} \ar[r] & \mathcal{O} \ar[r] & 0 } $$ which is the desired extension. We omit the verification that these constructions are mutually inverse.
Let $\mathcal{L}$ be an invertible $\mathcal{O}$-module. Consider the presheaf $\mathcal{L}^*$ defined by the rule $$ U \longmapsto \{s \in \mathcal{L}(U) \text{ such that } \mathcal{O}_U \xrightarrow{s \cdot -} \mathcal{L}_U \text{ is an isomorphism}\} $$ This presheaf satisfies the sheaf condition. Moreover, if $f \in \mathcal{O}^*(U)$ and $s \in \mathcal{L}^*(U)$, then clearly $fs \in \mathcal{L}^*(U)$. By the same token, if $s, s' \in \mathcal{L}^*(U)$ then there exists a unique $f \in \mathcal{O}^*(U)$ such that $fs = s'$. Moreover, the sheaf $\mathcal{L}^*$ has sections locally by Modules on Sites, Lemma \ref{sites-modules-lemma-invertible-is-locally-free-rank-1}. In other words we see that $\mathcal{L}^*$ is a $\mathcal{O}^*$-torsor. Thus we get a map $$ \begin{matrix} \text{set of invertible sheaves on }(\mathcal{C}, \mathcal{O}) \\ \text{ up to isomorphism} \end{matrix} \longrightarrow \begin{matrix} \text{set of }\mathcal{O}^*\text{-torsors} \\ \text{ up to isomorphism} \end{matrix} $$ We omit the verification that this is a homomorphism of abelian groups. By Lemma \ref{lemma-torsors-h1} the right hand side is canonically bijective to $H^1(\mathcal{C}, \mathcal{O}^*)$. Thus we have to show this map is injective and surjective. \medskip\noindent Injective. If the torsor $\mathcal{L}^*$ is trivial, this means by Lemma \ref{lemma-trivial-torsor} that $\mathcal{L}^*$ has a global section. Hence this means exactly that $\mathcal{L} \cong \mathcal{O}$ is the neutral element in $\Pic(\mathcal{O})$. \medskip\noindent Surjective. Let $\mathcal{F}$ be an $\mathcal{O}^*$-torsor. Consider the presheaf of sets $$ \mathcal{L}_1 : U \longmapsto (\mathcal{F}(U) \times \mathcal{O}(U))/\mathcal{O}^*(U) $$ where the action of $f \in \mathcal{O}^*(U)$ on $(s, g)$ is $(fs, f^{-1}g)$. Then $\mathcal{L}_1$ is a presheaf of $\mathcal{O}$-modules by setting $(s, g) + (s', g') = (s, g + (s'/s)g')$ where $s'/s$ is the local section $f$ of $\mathcal{O}^*$ such that $fs = s'$, and $h(s, g) = (s, hg)$ for $h$ a local section of $\mathcal{O}$. We omit the verification that the sheafification $\mathcal{L} = \mathcal{L}_1^\#$ is an invertible $\mathcal{O}$-module whose associated $\mathcal{O}^*$-torsor $\mathcal{L}^*$ is isomorphic to $\mathcal{F}$.
Recall that the functor $j_U^{-1}$ of restriction to $U$ is a right adjoint to the functor $j_{U!}$ of extension by $0$, see Modules on Sites, Section \ref{sites-modules-section-localize}. Moreover, $j_{U!}$ is exact. Hence (1) follows from Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}. \medskip\noindent By definition $H^p(U, \mathcal{F}) = H^p(\mathcal{I}^\bullet(U))$ where $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution in $\textit{Mod}(\mathcal{O})$. By the above we see that $\mathcal{F}|_U \to \mathcal{I}^\bullet|_U$ is an injective resolution in $\textit{Mod}(\mathcal{O}_U)$. Hence $H^p(U, \mathcal{F}|_U)$ is equal to $H^p(\mathcal{I}^\bullet|_U(U))$. Of course $\mathcal{F}(U) = \mathcal{F}|_U(U)$ for any sheaf $\mathcal{F}$ on $\mathcal{C}$. Hence the equality in (2).
Since it is clear that $\Gamma(\mathcal{C}, g^{-1}\mathcal{F}) = \Gamma(\mathcal{D}, \mathcal{F})$ by hypothesis (e), it suffices to show that $g^{-1}$ transforms injective abelian sheaves into injective abelian sheaves. As usual we use Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} to see this. The left adjoint to $g^{-1}$ is $g_! = f^{-1}$ with the notation of Sites, Lemma \ref{sites-lemma-bigger-site} which is an exact functor. Hence the lemma does indeed apply.
By Lemma \ref{lemma-cech-exact-presheaves} a short exact sequence of abelian presheaves $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is turned into a short exact sequence of complexes of $\mathbf{Z}$-modules. Hence we can use Homology, Lemma \ref{homology-lemma-long-exact-sequence-cochain} to get the boundary maps $\delta_{\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3} : \check{H}^n(\mathcal{U}, \mathcal{F}_3) \to \check{H}^{n + 1}(\mathcal{U}, \mathcal{F}_1)$ and a corresponding long exact sequence. We omit the verification that these maps are compatible with maps between short exact sequences of presheaves.
This is a tautology based on the fact that \begin{align*} \Hom_{\textit{PAb}(\mathcal{C})}( \bigoplus_{i_0 \ldots i_p} \mathbf{Z}_{U_{i_0} \times_U \ldots \times_U U_{i_p}}, \mathcal{F}) & = \prod_{i_0 \ldots i_p} \Hom_{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{U_{i_0} \times_U \ldots \times_U U_{i_p}}, \mathcal{F}) \\ & = \prod_{i_0 \ldots i_p} \mathcal{F}(U_{i_0} \times_U \ldots \times_U U_{i_p}) \end{align*} see Modules on Sites, Lemma \ref{sites-modules-lemma-obvious-adjointness}.
Let $V$ be an object of $\mathcal{C}$. In the proof of Lemma \ref{lemma-homology-complex} we saw that $\mathbf{Z}_{\mathcal{U}, \bullet}(V)$ is isomorphic as a complex to a direct sum of complexes which are homotopic to $\mathbf{Z}$ placed in degree zero. Hence also $\mathbf{Z}_{\mathcal{U}, \bullet}(V) \otimes_\mathbf{Z} \mathcal{O}(V)$ is isomorphic as a complex to a direct sum of complexes which are homotopic to $\mathcal{O}(V)$ placed in degree zero. Or you can use Modules on Sites, Lemma \ref{sites-modules-lemma-flat-resolution-of-flat}, which applies since the presheaves $\mathbf{Z}_{\mathcal{U}, i}$ are flat, and the proof of Lemma \ref{lemma-homology-complex} shows that $H_0(\mathbf{Z}_{\mathcal{U}, \bullet})$ is a flat presheaf also.
Apply Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} to the categories $\mathcal{A} = \textit{Ab}(\mathcal{C})$, $\mathcal{B} = \textit{PAb}(\mathcal{C})$, the inclusion functor and sheafification. (See Modules on Sites, Section \ref{sites-modules-section-abelian-sheaves} to see that all assumptions of the lemma are satisfied.)
By Lemma \ref{lemma-injective-abelian-sheaf-injective-presheaf} we see that $\mathcal{I}$ is an injective object in $\textit{PAb}(\mathcal{C})$. Hence we can apply Lemma \ref{lemma-cech-cohomology-derived-presheaves} (or its proof) to see the vanishing of higher {\v C}ech cohomology group. For the zeroth see Lemma \ref{lemma-cech-h0}.
Let $\mathcal{F}$ be an abelian sheaf. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. Consider the double complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)$ with terms $\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^q)$. Next, consider the associated total complex $\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))$, see Homology, Definition \ref{homology-definition-associated-simple-complex}. There is a map of complexes $$ \alpha : \Gamma(U, \mathcal{I}^\bullet) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)) $$ coming from the maps $\mathcal{I}^q(U) \to \check{H}^0(\mathcal{U}, \mathcal{I}^q)$ and a map of complexes $$ \beta : \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)) $$ coming from the map $\mathcal{F} \to \mathcal{I}^0$. We can apply Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution} to see that $\alpha$ is a quasi-isomorphism. Namely, Lemma \ref{lemma-injective-trivial-cech} implies that the $q$th row of the double complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)$ is a resolution of $\Gamma(U, \mathcal{I}^q)$. Hence $\alpha$ becomes invertible in $D^{+}(\mathbf{Z})$ and the transformation of the lemma is the composition of $\beta$ followed by the inverse of $\alpha$. We omit the verification that this is functorial.
To see this we construct an inverse map. Namely, let $\mathcal{F}$ be a $\mathcal{G}|_U$-torsor on $\mathcal{C}/U$ whose restriction to $\mathcal{C}/U_i$ is trivial. By Lemma \ref{lemma-trivial-torsor} this means there exists a section $s_i \in \mathcal{F}(U_i)$. On $U_{i_0} \times_U U_{i_1}$ there is a unique section $s_{i_0i_1}$ of $\mathcal{G}$ such that $s_{i_0i_1} \cdot s_{i_0}|_{U_{i_0} \times_U U_{i_1}} = s_{i_1}|_{U_{i_0} \times_U U_{i_1}}$. An easy computation shows that $s_{i_0i_1}$ is a {\v C}ech cocycle and that its class is well defined (i.e., does not depend on the choice of the sections $s_i$). The inverse maps the isomorphism class of $\mathcal{F}$ to the cohomology class of the cocycle $(s_{i_0i_1})$. We omit the verification that this map is indeed an inverse.
This is a Grothendieck spectral sequence (see Derived Categories, Lemma \ref{derived-lemma-grothendieck-spectral-sequence}) for the functors $$ i :  \textit{Ab}(\mathcal{C}) \to \textit{PAb}(\mathcal{C}) \quad\text{and}\quad \check{H}^0(\mathcal{U}, - ) : \textit{PAb}(\mathcal{C}) \to \textit{Ab}. $$ Namely, we have $\check{H}^0(\mathcal{U}, i(\mathcal{F})) = \mathcal{F}(U)$ by Lemma \ref{lemma-cech-h0}. We have that $i(\mathcal{I})$ is {\v C}ech acyclic by Lemma \ref{lemma-injective-trivial-cech}. And we have that $\check{H}^p(\mathcal{U}, -) = R^p\check{H}^0(\mathcal{U}, -)$ as functors on $\textit{PAb}(\mathcal{C})$ by Lemma \ref{lemma-cech-cohomology-derived-presheaves}. Putting everything together gives the lemma.
Let $\mathcal{F}$ and $\text{Cov}$ be as in the lemma. We will indicate this by saying ``$\mathcal{F}$ has vanishing higher {\v C}ech cohomology for any $\mathcal{U} \in \text{Cov}$''. Choose an embedding $\mathcal{F} \to \mathcal{I}$ into an injective abelian sheaf. By Lemma \ref{lemma-injective-trivial-cech} $\mathcal{I}$ has vanishing higher {\v C}ech cohomology for any $\mathcal{U} \in \text{Cov}$. Let $\mathcal{Q} = \mathcal{I}/\mathcal{F}$ so that we have a short exact sequence $$ 0 \to \mathcal{F} \to \mathcal{I} \to \mathcal{Q} \to 0. $$ By Lemma \ref{lemma-ses-cech-h1} and our assumption (2) this sequence gives rise to an exact sequence $$ 0 \to \mathcal{F}(U) \to \mathcal{I}(U) \to \mathcal{Q}(U) \to 0. $$ for every $U \in \mathcal{B}$. Hence for any $\mathcal{U} \in \text{Cov}$ we get a short exact sequence of {\v C}ech complexes $$ 0 \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{Q}) \to 0 $$ since each term in the {\v C}ech complex is made up out of a product of values over elements of $\mathcal{B}$ by assumption (1). In particular we have a long exact sequence of {\v C}ech cohomology groups for any covering $\mathcal{U} \in \text{Cov}$. This implies that $\mathcal{Q}$ is also an abelian sheaf with vanishing higher {\v C}ech cohomology for all $\mathcal{U} \in \text{Cov}$. \medskip\noindent Next, we look at the long exact cohomology sequence $$ \xymatrix{ 0 \ar[r] & H^0(U, \mathcal{F}) \ar[r] & H^0(U, \mathcal{I}) \ar[r] & H^0(U, \mathcal{Q}) \ar[lld] \\ & H^1(U, \mathcal{F}) \ar[r] & H^1(U, \mathcal{I}) \ar[r] & H^1(U, \mathcal{Q}) \ar[lld] \\ & \ldots & \ldots & \ldots \\ } $$ for any $U \in \mathcal{B}$. Since $\mathcal{I}$ is injective we have $H^n(U, \mathcal{I}) = 0$ for $n > 0$ (see Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}). By the above we see that $H^0(U, \mathcal{I}) \to H^0(U, \mathcal{Q})$ is surjective and hence $H^1(U, \mathcal{F}) = 0$. Since $\mathcal{F}$ was an arbitrary abelian sheaf with vanishing higher {\v C}ech cohomology for all $\mathcal{U} \in \text{Cov}$ we conclude that also $H^1(U, \mathcal{Q}) = 0$ since $\mathcal{Q}$ is another of these sheaves (see above). By the long exact sequence this in turn implies that $H^2(U, \mathcal{F}) = 0$. And so on and so forth.
By Stacks, Definition \ref{stacks-definition-gerbe} there exists a covering $\mathcal{U} = \{U_i \to U\}$ and $x_i$ in $\mathcal{S}$ lying over $U_i$. Write $U_{ij} = U_i \times_U U_j$. By (1) after refining the covering we may assume that $H^1(U_i, \mathcal{G}) = 0$ and $H^1(U_{ij}, \mathcal{G}) = 0$. Consider the sheaf $$ \mathcal{F}_{ij} = \mathit{Isom}(x_i|_{U_{ij}}, x_j|_{U_{ij}}) $$ on $\mathcal{C}/U_{ij}$. Since $\mathcal{G}|_{U_{ij}} = \mathit{Aut}(x_i|_{U_{ij}})$ we see that there is an action $$ \mathcal{G}|_{U_{ij}} \times \mathcal{F}_{ij} \to \mathcal{F}_{ij} $$ by precomposition. It is clear that $\mathcal{F}_{ij}$ is a pseudo $\mathcal{G}|_{U_{ij}}$-torsor and in fact a torsor because any two objects of a gerbe are locally isomorphic. By our choice of the covering and by Lemma \ref{lemma-torsors-h1} these torsors are trivial (and hence have global sections by Lemma \ref{lemma-trivial-torsor}). In other words, we can choose isomorphisms $$ \varphi_{ij} : x_i|_{U_{ij}} \longrightarrow x_j|_{U_{ij}} $$ To find an object $x$ over $U$ we are going to massage our choice of these $\varphi_{ij}$ to get a descent datum (which is necessarily effective as $p : \mathcal{S} \to \mathcal{C}$ is a stack). Namely, the obstruction to being a descent datum is that the cocycle condition may not hold. Namely, set $U_{ijk} = U_i \times_U U_j \times_U U_k$. Then we can consider $$ g_{ijk} = \varphi_{ik}^{-1}|_{U_{ijk}} \circ \varphi_{jk}|_{U_{ijk}} \circ \varphi_{ij}|_{U_{ijk}} $$ which is an automorphism of $x_i$ over $U_{ijk}$. Thus we may and do consider $g_{ijk}$ as a section of $\mathcal{G}$ over $U_{ijk}$. A computation (omitted) shows that $(g_{i_0i_1i_2})$ is a $2$-cocycle in the {\v C}ech complex ${\check C}^\bullet(\mathcal{U}, \mathcal{G})$ of $\mathcal{G}$ with respect to the covering $\mathcal{U}$. By the spectral sequence of Lemma \ref{lemma-cech-spectral-sequence} and since $H^1(U_i, \mathcal{G}) = 0$ for all $i$ we see that ${\check H}^2(\mathcal{U}, \mathcal{G}) \to H^2(U, \mathcal{G})$ is injective. Hence $(g_{i_0i_1i_2})$ is a coboundary by our assumption that $H^2(U, \mathcal{G}) = 0$. Thus we can find sections $g_{ij} \in \mathcal{G}(U_{ij})$ such that $g_{ik}^{-1}|_{U_{ijk}} g_{jk}|_{U_{ijk}}  g_{ij}|_{U_{ijk}} = g_{ijk}$ for all $i, j, k$. After replacing $\varphi_{ij}$ by $\varphi_{ij}g_{ij}^{-1}$ we see that $\varphi_{ij}$ gives a descent datum on the objects $x_i$ over $U_i$ and the proof is complete.
Apply Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} to the categories $\mathcal{A} = \textit{Mod}(\mathcal{O})$, $\mathcal{B} = \textit{PMod}(\mathcal{O})$, the inclusion functor and sheafification. (See Modules on Sites, Section \ref{sites-modules-section-sheafification-presheaves-modules} to see that all assumptions of the lemma are satisfied.)
Lemma \ref{lemma-cech-map-into} gives the first equality in the following sequence of equalities \begin{align*} \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}) & = \Mor_{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{\mathcal{U}, \bullet}, \mathcal{I}) \\ & = \Mor_{\textit{PMod}(\mathbf{Z})}( \mathbf{Z}_{\mathcal{U}, \bullet}, \mathcal{I}) \\ & = \Mor_{\textit{PMod}(\mathcal{O})}( \mathbf{Z}_{\mathcal{U}, \bullet} \otimes_{p, \mathbf{Z}} \mathcal{O}, \mathcal{I}) \end{align*} The third equality by Modules on Sites, Lemma \ref{sites-modules-lemma-adjointness-tensor-restrict-presheaves}. By Lemma \ref{lemma-injective-module-injective-presheaf} we see that $\mathcal{I}$ is an injective object in $\textit{PMod}(\mathcal{O})$. Hence $\Hom_{\textit{PMod}(\mathcal{O})}(-, \mathcal{I})$ is an exact functor. By Lemma \ref{lemma-complex-tensored-still-exact} we see the vanishing of higher {\v C}ech cohomology groups. For the zeroth see Lemma \ref{lemma-cech-h0}.
By Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors} the $\delta$-functor $(\mathcal{F} \mapsto H^p(U, \mathcal{F}))_{p \geq 0}$ is universal. The functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$, $\mathcal{F} \mapsto \mathcal{F}_{ab}$ is exact. Hence $(\mathcal{F} \mapsto H^p(U, \mathcal{F}_{ab}))_{p \geq 0}$ is a $\delta$-functor also. Suppose we show that $(\mathcal{F} \mapsto H^p(U, \mathcal{F}_{ab}))_{p \geq 0}$ is also universal. This will imply the second statement of the lemma by uniqueness of universal $\delta$-functors, see Homology, Lemma \ref{homology-lemma-uniqueness-universal-delta-functor}. Since $\textit{Mod}(\mathcal{O})$ has enough injectives, it suffices to show that $H^i(U, \mathcal{I}_{ab}) = 0$ for any injective object $\mathcal{I}$ in $\textit{Mod}(\mathcal{O})$, see Homology, Lemma \ref{homology-lemma-efface-implies-universal}. \medskip\noindent Let $\mathcal{I}$ be an injective object of $\textit{Mod}(\mathcal{O})$. Apply Lemma \ref{lemma-cech-vanish-collection} with $\mathcal{F} = \mathcal{I}$, $\mathcal{B} = \mathcal{C}$ and $\text{Cov} = \text{Cov}_\mathcal{C}$. Assumption (3) of that lemma holds by Lemma \ref{lemma-injective-module-trivial-cech}. Hence we see that $H^i(U, \mathcal{I}_{ab}) = 0$ for every object $U$ of $\mathcal{C}$. \medskip\noindent If $\mathcal{C}$ has a final object then this also implies the first equality. If not, then according to Sites, Lemma \ref{sites-lemma-topos-good-site} we see that the ringed topos $(\Sh(\mathcal{C}), \mathcal{O})$ is equivalent to a ringed topos where the underlying site does have a final object. Hence the lemma follows.
The statement for $p = 0$ is true because the product of sheaves is equal to the product of the underlying presheaves, see Sites, Lemma \ref{sites-lemma-limit-sheaf}. Proof for $p = 1$. Set $\mathcal{F} = \prod \mathcal{F}_i$. Let $\xi \in H^1(U, \mathcal{F})$ map to zero in $\prod H^1(U, \mathcal{F}_i)$. By locality of cohomology, see Lemma \ref{lemma-kill-cohomology-class-on-covering}, there exists a covering $\mathcal{U} = \{U_j \to U\}$ such that $\xi|_{U_j} = 0$ for all $j$. By Lemma \ref{lemma-cech-h1} this means $\xi$ comes from an element $\check \xi \in \check H^1(\mathcal{U}, \mathcal{F})$. Since the maps $\check H^1(\mathcal{U}, \mathcal{F}_i) \to H^1(U, \mathcal{F}_i)$ are injective for all $i$ (by Lemma \ref{lemma-cech-h1}), and since the image of $\xi$ is zero in $\prod H^1(U, \mathcal{F}_i)$ we see that the image $\check \xi_i = 0$ in $\check H^1(\mathcal{U}, \mathcal{F}_i)$. However, since $\mathcal{F} = \prod \mathcal{F}_i$ we see that $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ is the product of the complexes $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}_i)$, hence by Homology, Lemma \ref{homology-lemma-product-abelian-groups-exact} we conclude that $\check \xi = 0$ as desired.
Let $j : \mathcal{C}/U \to \mathcal{C}$ and $j' : \mathcal{C}/U' \to \mathcal{C}$ be the localization morphisms (Modules on Sites, Section \ref{sites-modules-section-localize}). Since $j_!$ is a left adjoint to restriction we see that for any sheaf $\mathcal{F}$ of $\mathcal{O}$-modules $$ \Hom_\mathcal{O}(j_!\mathcal{O}_U, \mathcal{F}) = \Hom_{\mathcal{O}_U}(\mathcal{O}_U, \mathcal{F}|_U) = \mathcal{F}(U) $$ Similarly, the sheaf $j'_!\mathcal{O}_{U'}$ represents the functor $\mathcal{F} \mapsto \mathcal{F}(U')$. Moreover below we describe a canonical map of $\mathcal{O}$-modules $$ j'_!\mathcal{O}_{U'} \longrightarrow j_!\mathcal{O}_U $$ which corresponds to the restriction mapping $\mathcal{F}(U) \to \mathcal{F}(U')$ via Yoneda's lemma (Categories, Lemma \ref{categories-lemma-yoneda}). It suffices to prove the displayed map of modules is injective, see Homology, Lemma \ref{homology-lemma-characterize-injectives}. \medskip\noindent To construct our map it suffices to construct a map between the presheaves which assign to an object $V$ of $\mathcal{C}$ the $\mathcal{O}(V)$-module $$ \bigoplus\nolimits_{\varphi' \in \Mor_\mathcal{C}(V, U')} \mathcal{O}(V) \quad\text{and}\quad \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{O}(V) $$ see Modules on Sites, Lemma \ref{sites-modules-lemma-extension-by-zero}. We take the map which maps the summand corresponding to $\varphi'$ to the summand corresponding to $\varphi = a \circ \varphi'$ by the identity map on $\mathcal{O}(V)$. As $a$ is a monomorphism, this map is injective. As sheafification is exact, the result follows.