hcju/stacksqa huggingface dataset:
Let $\mathcal{I}^\bullet$ be a bounded below complex of injectives representing $K$. Consider the double complex with terms $$ A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p) $$ where the horizontal arrows come from Lemma \ref{lemma-cech-complex} and the vertical arrows from the differentials of the complex $\mathcal{I}^\bullet$. The rows of the double complex are exact in positive degrees and evaluate to $\Gamma(\mathcal{C}_{total}, \mathcal{I}^q)$ in degree $0$. On the other hand, since restriction to $\mathcal{C}_p$ is exact (Lemma \ref{lemma-restriction-to-components-site}) the complex $\mathcal{I}_p^\bullet$ represents $K_p$ in $D(\mathcal{C}_p)$. The sheaves $\mathcal{I}_p^q$ are injective abelian sheaves on $\mathcal{C}_p$ (Lemma \ref{lemma-restriction-injective-to-component-site}). Hence the cohomology of the columns computes the groups $H^q(\mathcal{C}_p, K_p)$. We conclude by applying Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and \ref{homology-lemma-double-complex-gives-resolution}.
Observe that ``$U$ viewed as object of $\mathcal{C}_{total}$'' is explained by the construction of $\mathcal{C}_{total}$ in Lemma \ref{lemma-simplicial-site-site} in case (A) and Lemma \ref{lemma-simplicial-cocontinuous-site} in case (B). The equality then follows from Lemma \ref{lemma-restriction-injective-to-component-site} and the definition of cohomology.
We encourage the reader to read the proof of Lemma \ref{lemma-simplicial-resolution-Z-site} first. We will use Lemma \ref{lemma-augmentation-site} and the description of the functors $g_{n!}$ in Lemma \ref{lemma-restriction-to-components-site} without further mention. In particular $g_{n!}(a_n^{-1}\mathcal{G})$ is the sheaf on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_m$ is the sheaf $$ \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}a_n^{-1}\mathcal{G} = \bigoplus\nolimits_{\varphi : [n] \to [m]} a_m^{-1}\mathcal{G} $$ As maps of the complex we take $\sum (-1)^i d^n_i$ where $d^n_i : g_{n!}(a_n^{-1}\mathcal{G}) \to g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})$ is the adjoint to the map $a_n^{-1}\mathcal{G} \to \bigoplus_{[n - 1] \to [n]} a_n^{-1}\mathcal{G} = g_n^{-1}g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})$ corresponding to the factor labeled with $\delta^n_i : [n - 1] \to [n]$. The map $g_{0!}(a_0^{-1}\mathcal{G}) \to a^{-1}\mathcal{G}$ is adjoint to the identity map of $a_0^{-1}\mathcal{G}$. Then $g_m^{-1}$ applied to the chain complex in degrees $\ldots, 2, 1, 0$ gives the complex $$ \ldots \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} a_m^{-1}\mathcal{G} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} a_m^{-1}\mathcal{G} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} a_m^{-1}\mathcal{G} $$ on $\mathcal{C}_m$. This is equal to $a_m^{-1}\mathcal{G}$ tensored over the constant sheaf $\mathbf{Z}$ with the complex $$ \ldots \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} \mathbf{Z} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} \mathbf{Z} \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} \mathbf{Z} $$ discussed in the proof of Lemma \ref{lemma-simplicial-resolution-Z-site}. There we have seen that this complex is homotopy equivalent to $\mathbf{Z}$ placed in degree $0$ which finishes the proof.
To construct the complex, by the Yoneda lemma, it suffices for any abelian sheaf $\mathcal{G}$ on $\mathcal{D}$ to construct a complex $$ 0 \to \Hom(\mathcal{G}, a_*\mathcal{F}) \to \Hom(\mathcal{G}, a_{0, *}\mathcal{F}_0) \to \Hom(\mathcal{G}, a_{1, *}\mathcal{F}_1) \to \ldots $$ functorially in $\mathcal{G}$. To do this apply $\Hom(-, \mathcal{F})$ to the exact complex of Lemma \ref{lemma-simplicial-resolution-augmentation} and use adjointness of pullback and pushforward. The exactness properties in degrees $-1, 0$ follow from the construction as $\Hom(-, \mathcal{F})$ is left exact. If $\mathcal{F}$ is an injective abelian sheaf, then the complex is exact because $\Hom(-, \mathcal{F})$ is exact.
Let $\mathcal{I}^\bullet$ be a bounded below complex of injectives representing $K$. Consider the double complex with terms $$ A^{p, q} = a_{p, *}\mathcal{I}^q_p $$ where the horizontal arrows come from Lemma \ref{lemma-augmentation-cech-complex} and the vertical arrows from the differentials of the complex $\mathcal{I}^\bullet$. The rows of the double complex are exact in positive degrees and evaluate to $a_*\mathcal{I}^q$ in degree $0$. On the other hand, since restriction to $\mathcal{C}_p$ is exact (Lemma \ref{lemma-restriction-to-components-site}) the complex $\mathcal{I}_p^\bullet$ represents $K_p$ in $D(\mathcal{C}_p)$. The sheaves $\mathcal{I}_p^q$ are injective abelian sheaves on $\mathcal{C}_p$ (Lemma \ref{lemma-restriction-injective-to-component-site}). Hence the cohomology of the columns computes $R^qa_{p, *}K_p$. We conclude by applying Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and \ref{homology-lemma-double-complex-gives-resolution}.
We will use the description of $g_{n!}$ given in Lemma \ref{lemma-restriction-to-components-site}. As maps of the complex we take $\sum (-1)^i d^n_i$ where $d^n_i : g_{n!}\mathcal{O}_n \to g_{n - 1!}\mathcal{O}_{n - 1}$ is the adjoint to the map $\mathcal{O}_n \to \bigoplus_{[n - 1] \to [n]} \mathcal{O}_n = g_n^*g_{n - 1!}\mathcal{O}_{n - 1}$ corresponding to the factor labeled with $\delta^n_i : [n - 1] \to [n]$. Then $g_m^{-1}$ applied to the complex gives the complex $$ \ldots \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} \mathcal{O}_m \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} \mathcal{O}_m \to \bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} \mathcal{O}_m $$ on $\mathcal{C}_m$. In other words, this is the complex associated to the free $\mathcal{O}_m$-module 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 $\mathcal{O}_m$ in degree $0$.
Observe that $\Hom(\mathcal{O}, \mathcal{F}) = \Gamma(\mathcal{C}_{total}, \mathcal{F})$ and $\Hom(g_{n!}\mathcal{O}_n, \mathcal{F}) = \Gamma(\mathcal{C}_n, \mathcal{F}_n)$. Hence this lemma is an immediate consequence of Lemma \ref{lemma-simplicial-resolution-ringed} and the fact that $\Hom(-, \mathcal{F})$ is exact if $\mathcal{F}$ is injective.
Let $\mathcal{I}^\bullet$ be a bounded below complex of injective $\mathcal{O}$-modules representing $K$. Consider the double complex with terms $$ A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p) $$ where the horizontal arrows come from Lemma \ref{lemma-cech-complex-modules} and the vertical arrows from the differentials of the complex $\mathcal{I}^\bullet$. Observe that $\Gamma(\mathcal{D}, -) = \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{O}_\mathcal{D}, -)$ on $\textit{Mod}(\mathcal{O}_\mathcal{D})$. Hence the lemma says rows of the double complex are exact in positive degrees and evaluate to $\Gamma(\mathcal{C}_{total}, \mathcal{I}^q)$ in degree $0$. Thus the total complex associated to the double complex computes $R\Gamma(\mathcal{C}_{total}, K)$ by Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}. On the other hand, since restriction to $\mathcal{C}_p$ is exact (Lemma \ref{lemma-restriction-to-components-site}) the complex $\mathcal{I}_p^\bullet$ represents $K_p$ in $D(\mathcal{C}_p)$. The sheaves $\mathcal{I}_p^q$ are totally acyclic on $\mathcal{C}_p$ (Lemma \ref{lemma-restriction-injective-to-component-limp}). Hence the cohomology of the columns computes the groups $H^q(\mathcal{C}_p, K_p)$ by Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-limp-acyclic}. We conclude by applying Homology, Lemma \ref{homology-lemma-first-quadrant-ss}.
Observe that ``$U$ viewed as object of $\mathcal{C}_{total}$'' is explained by the construction of $\mathcal{C}_{total}$ in Lemma \ref{lemma-simplicial-site-site} in case (A) and Lemma \ref{lemma-simplicial-cocontinuous-site} in case (B). In both cases the functor $\mathcal{C}_n \to \mathcal{C}$ is continuous and cocontinuous, see Lemma \ref{lemma-restriction-to-components-site}, and $g_n^{-1}\mathcal{O} = \mathcal{O}_n$ by definition. Hence the result is a special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-same-cohomology}.
Observe that $a^*\mathcal{G}$ is the $\mathcal{O}$-module on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_m$ is the $\mathcal{O}_m$-module $a_m^*\mathcal{G}$. The description of the functors $g_{n!}$ on modules in Lemma \ref{lemma-restriction-module-to-components-site} shows that $g_{n!}(a_n^*\mathcal{G})$ is the $\mathcal{O}$-module on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_m$ is the $\mathcal{O}_m$-module $$ \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*a_n^*\mathcal{G} = \bigoplus\nolimits_{\varphi : [n] \to [m]} a_m^*\mathcal{G} $$ The rest of the proof is exactly the same as the proof of Lemma \ref{lemma-simplicial-resolution-augmentation}, replacing $a_m^{-1}\mathcal{G}$ by $a_m^*\mathcal{G}$.
To construct the complex, by the Yoneda lemma, it suffices for any $\mathcal{O}_\mathcal{D}$-modules $\mathcal{G}$ on $\mathcal{D}$ to construct a complex $$ 0 \to \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, a_*\mathcal{F}) \to \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, a_{0, *}\mathcal{F}_0) \to \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, a_{1, *}\mathcal{F}_1) \to \ldots $$ functorially in $\mathcal{G}$. To do this apply $\Hom_\mathcal{O}(-, \mathcal{F})$ to the exact complex of Lemma \ref{lemma-simplicial-resolution-augmentation-modules} and use adjointness of pullback and pushforward. The exactness properties in degrees $-1, 0$ follow from the construction as $\Hom_\mathcal{O}(-, \mathcal{F})$ is left exact. If $\mathcal{F}$ is an injective $\mathcal{O}$-module, then the complex is exact because $\Hom_\mathcal{O}(-, \mathcal{F})$ is exact.
Let $\mathcal{I}^\bullet$ be a bounded below complex of injective $\mathcal{O}$-modules representing $K$. Consider the double complex with terms $$ A^{p, q} = a_{p, *}\mathcal{I}^q_p $$ where the horizontal arrows come from Lemma \ref{lemma-augmentation-cech-complex-modules} and the vertical arrows from the differentials of the complex $\mathcal{I}^\bullet$. The lemma says rows of the double complex are exact in positive degrees and evaluate to $a_*\mathcal{I}^q$ in degree $0$. Thus the total complex associated to the double complex computes $Ra_*K$ by Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}. On the other hand, since restriction to $\mathcal{C}_p$ is exact (Lemma \ref{lemma-restriction-to-components-site}) the complex $\mathcal{I}_p^\bullet$ represents $K_p$ in $D(\mathcal{C}_p)$. The sheaves $\mathcal{I}_p^q$ are totally acyclic on $\mathcal{C}_p$ (Lemma \ref{lemma-restriction-injective-to-component-limp}). Hence the cohomology of the columns are the sheaves $R^qa_{p, *}K_p$ by Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-limp-acyclic}. We conclude by applying Homology, Lemma \ref{homology-lemma-first-quadrant-ss}.
In each case the key is that the pullback functors compose to pullback functor; for part (4) see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pullback-composition}. We show how the argument works in case (1) and omit the proof in the other cases. The category $\Delta$ is generated by the morphisms the morphisms $\delta^n_j$ and $\sigma^n_j$, see Simplicial, Lemma \ref{simplicial-lemma-face-degeneracy}. Hence we only need to check the maps $(f_{\delta^n_j})^{-1}\mathcal{F}_{n - 1} \to \mathcal{F}_n$ and $(f_{\sigma^n_j})^{-1}\mathcal{F}_{n + 1} \to \mathcal{F}_n$ are isomorphisms, see Simplicial, Lemma \ref{simplicial-lemma-characterize-simplicial-object} for notation. Since $\sigma^n_j \circ \delta_j^{n + 1} = \text{id}_{[n]}$ the composition $$ \mathcal{F}_n = (f_{\sigma^n_j})^{-1} (f_{\delta_j^{n + 1}})^{-1} \mathcal{F}_n \to (f_{\sigma^n_j})^{-1} \mathcal{F}_{n + 1} \to \mathcal{F}_n $$ is the identity. Thus the result for $\delta^{n + 1}_j$ implies the result for $\sigma^n_j$.
We abbreviate $d^n_j = f_{\delta^n_j} : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{n - 1})$. The condition on $\alpha$ in the statement of the lemma makes sense because $$ d^1_1 \circ d^2_2 = d^1_1 \circ d^2_1, \quad d^1_1 \circ d^2_0 = d^1_0 \circ d^2_2, \quad d^1_0 \circ d^2_0 = d^1_0 \circ d^2_1 $$ as morphisms of topoi $\Sh(\mathcal{C}_2) \to \Sh(\mathcal{C}_0)$, see Simplicial, Remark \ref{simplicial-remark-relations}. Hence we can picture these maps as follows $$ \xymatrix{ & (d^2_0)^{-1}(d^1_1)^{-1}\mathcal{F} \ar[r]_-{(d^2_0)^{-1}\alpha} & (d^2_0)^{-1}(d^1_0)^{-1}\mathcal{F} \ar@{=}[rd] & \\ (d^2_2)^{-1}(d^1_0)^{-1}\mathcal{F} \ar@{=}[ru] & & & (d^2_1)^{-1}(d^1_0)^{-1}\mathcal{F} \\ & (d^2_2)^{-1}(d^1_1)^{-1}\mathcal{F} \ar[lu]^{(d^2_2)^{-1}\alpha} \ar@{=}[r] & (d^2_1)^{-1}(d^1_1)^{-1}\mathcal{F} \ar[ru]_{(d^2_1)^{-1}\alpha} } $$ and the condition signifies the diagram is commutative. It is clear that given a cartesian sheaf $\mathcal{G}$ of sets (resp.\ abelian groups) on $\mathcal{C}_{total}$ we can set $\mathcal{F} = \mathcal{G}_0$ and $\alpha$ equal to the composition $$ (d_1^1)^{-1}\mathcal{G}_0 \to \mathcal{G}_1 \leftarrow (d_1^0)^{-1}\mathcal{G}_0 $$ where the arrows are invertible as $\mathcal{G}$ is cartesian. To prove this functor is an equivalence we construct a quasi-inverse. The construction of the quasi-inverse is analogous to the construction discussed in Descent, Section \ref{descent-section-descent-modules} from which we borrow the notation $\tau^n_i : [0] \to [n]$, $0 \mapsto i$ and $\tau^n_{ij} : [1] \to [n]$, $0 \mapsto i$, $1 \mapsto j$. Namely, given a pair $(\mathcal{F}, \alpha)$ as in the lemma we set $\mathcal{G}_n = (f_{\tau^n_n})^{-1}\mathcal{F}$. Given $\varphi : [n] \to [m]$ we define $\mathcal{G}(\varphi) : (f_\varphi)^{-1}\mathcal{G}_n \to \mathcal{G}_m$ using $$ \xymatrix{ (f_\varphi)^{-1}\mathcal{G}_n \ar@{=}[r] & (f_\varphi)^{-1}(f_{\tau^n_n})^{-1}\mathcal{F} \ar@{=}[r] & (f_{\tau^m_{\varphi(n)}})^{-1}\mathcal{F} \ar@{=}[r] & (f_{\tau^m_{\varphi(n)m}})^{-1}(d^1_1)^{-1}\mathcal{F} \ar[d]^{(f_{\tau^m_{\varphi(n)m}})^{-1}\alpha} \\ & \mathcal{G}_m \ar@{=}[r] & (f_{\tau^m_m})^{-1}\mathcal{F} \ar@{=}[r] & (f_{\tau^m_{\varphi(n)m}})^{-1}(d^1_0)^{-1}\mathcal{F} } $$ We omit the verification that the commutativity of the displayed diagram above implies the maps compose correctly and hence give rise to a sheaf on $\mathcal{C}_{total}$, see Lemma \ref{lemma-describe-sheaves-simplicial-site-site}. We also omit the verification that the two functors are quasi-inverse to each other.
To see we obtain a weak Serre subcategory in (1) we check the conditions listed in Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}. First, if $\varphi : \mathcal{F} \to \mathcal{G}$ is a map between cartesian abelian sheaves, then $\Ker(\varphi)$ and $\Coker(\varphi)$ are cartesian too because the restriction functors $\Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}_n)$ and the functors $f_\varphi^{-1}$ are exact. Similarly, if $$ 0 \to \mathcal{F} \to \mathcal{H} \to \mathcal{G} \to 0 $$ is a short exact sequence of abelian sheaves on $\mathcal{C}_{total}$ with $\mathcal{F}$ and $\mathcal{G}$ cartesian, then it follows that $\mathcal{H}$ is cartesian from the 5-lemma. To see the property of colimits, use that colimits commute with pullback as pullback is a left adjoint. In the case of modules we argue in the same manner, using the exactness of flat pullback (Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact}) and the fact that it suffices to check the condition for $f_{\delta^n_j}$, see Lemma \ref{lemma-check-cartesian-module}.
Proof of (1). Since $g_{n!}$ is exact, it induces a functor on derived categories adjoint to $g_n^{-1}$. The map is the adjoint of the map $K_n \to (g_n^{-1}g_{n!}\mathbf{Z}) \otimes^\mathbf{L}_\mathbf{Z} K_n$ corresponding to $\mathbf{Z} \to g_n^{-1}g_{n!}\mathbf{Z}$ which in turn is adjoint to $\text{id} : g_{n!}\mathbf{Z} \to g_{n!}\mathbf{Z}$. Using the description of $g_{n!}$ given in Lemma \ref{lemma-restriction-to-components-site} we see that the restriction to $\mathcal{C}_m$ of this map is $$ \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}K_n \longrightarrow \bigoplus\nolimits_{\varphi : [n] \to [m]} K_m $$ Thus the statement is clear. \medskip\noindent Proof of (2). Since $g_{n!}$ is exact (Lemma \ref{lemma-exactness-g-shriek-modules}), it induces a functor on derived categories adjoint to $g_n^*$ (also exact). The map is the adjoint of the map $K_n \to (g_n^*g_{n!}\mathcal{O}_n) \otimes^\mathbf{L}_{\mathcal{O}_n} K_n$ corresponding to $\mathcal{O}_n \to g_n^*g_{n!}\mathcal{O}_n$ which in turn is adjoint to $\text{id} : g_{n!}\mathcal{O}_n \to g_{n!}\mathcal{O}_n$. Using the description of $g_{n!}$ given in Lemma \ref{lemma-restriction-module-to-components-site} we see that the restriction to $\mathcal{C}_m$ of this map is $$ \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*K_n \longrightarrow \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{O}_n \otimes_{\mathcal{O}_m} K_m = \bigoplus\nolimits_{\varphi : [n] \to [m]} K_m $$ Thus the statement is clear.
Assume $\mathcal{F}$ is quasi-coherent. Since pullbacks of quasi-coherent modules are quasi-coherent (Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}) we see that $\mathcal{F}_n$ is a quasi-coherent $\mathcal{O}_n$-module for all $n$. To show that $\mathcal{F}$ is cartesian, let $U$ be an object of $\mathcal{C}_n$ for some $n$. Let us view $U$ as an object of $\mathcal{C}_{total}$. Because $\mathcal{F}$ is quasi-coherent there exists a covering $\{U_i \to U\}$ and for each $i$ a presentation $$ \bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to \mathcal{F}|_{\mathcal{C}_{total}/U_i} \to 0 $$ Observe that $\{U_i \to U\}$ is a covering of $\mathcal{C}_n$ by the construction of the site $\mathcal{C}_{total}$. Next, let $V$ be an object of $\mathcal{C}_m$ for some $m$ and let $V \to U$ be a morphism of $\mathcal{C}_{total}$ lying over $\varphi : [n] \to [m]$. The fibre products $V_i = V \times_U U_i$ exist and we get an induced covering $\{V_i \to V\}$ in $\mathcal{C}_m$. Restricting the presentation above to the sites $\mathcal{C}_n/U_i$ and $\mathcal{C}_m/V_i$ we obtain presentations $$ \bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to \mathcal{F}_n|_{\mathcal{C}_n/U_i} \to 0 $$ and $$ \bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_m/V_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_m/V_i} \to \mathcal{F}_m|_{\mathcal{C}_m/V_i} \to 0 $$ These presentations are compatible with the map $\mathcal{F}(\varphi) : f_\varphi^*\mathcal{F}_n \to \mathcal{F}_m$ (as this map is defined using the restriction maps of $\mathcal{F}$ along morphisms of $\mathcal{C}_{total}$ lying over $\varphi$). We conclude that $\mathcal{F}(\varphi)|_{\mathcal{C}_m/V_i}$ is an isomorphism. As $\{V_i \to V\}$ is a covering we conclude $\mathcal{F}(\varphi)|_{\mathcal{C}_m/V}$ is an isomorphism. Since $V$ and $U$ were arbitrary this proves that $\mathcal{F}$ is cartesian. (In case A use Sites, Lemma \ref{sites-lemma-morphism-of-sites-covering}.) \medskip\noindent Conversely, assume $\mathcal{F}_n$ is quasi-coherent for all $n$ and that $\mathcal{F}$ is cartesian. Then for any $n$ and object $U$ of $\mathcal{C}_n$ we can choose a covering $\{U_i \to U\}$ of $\mathcal{C}_n$ and for each $i$ a presentation $$ \bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to \mathcal{F}_n|_{\mathcal{C}_n/U_i} \to 0 $$ Pulling back to $\mathcal{C}_{total}/U_i$ we obtain complexes $$ \bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to \mathcal{F}|_{\mathcal{C}_{total}/U_i} \to 0 $$ of modules on $\mathcal{C}_{total}/U_i$. Then the property that $\mathcal{F}$ is cartesian implies that this is exact. We omit the details.
First, if $K = \mathbf{Z}$, then this is the construction of Derived Categories, Example \ref{derived-example-key-postnikov} applied to the complex $$ \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z} $$ in $\textit{Ab}(\mathcal{C}_{total})$ combined with the fact that this complex represents $K = \mathbf{Z}$ in $D(\mathcal{C}_{total})$ by Lemma \ref{lemma-simplicial-resolution-Z-site}. The general case follows from this, the fact that the exact functor $- \otimes^\mathbf{L}_\mathbf{Z} K$ sends Postnikov systems to Postnikov systems, and that $- \otimes^\mathbf{L}_\mathbf{Z} K$ commutes with homotopy colimits.
Consider the objects $X_n$ and the Postnikov system $Y_n$ associated to $K$ in Lemma \ref{lemma-abelian-postnikov}. As $K = \text{hocolim} Y_n[n]$ the map $K \to K'$ induces a compatible family of morphisms $Y_n[n] \to K'$. By (1) and Lemma \ref{lemma-derived-cartesian-shriek} we have $X_n = g_{n!}K_n$. Since $Y_0 = X_0$ we find that $K_0 \to K'_0$ being zero implies $Y_0 \to K'$ is zero. Suppose we've shown that the map $Y_n[n] \to K'$ is zero for some $n \geq 0$. From the distinguished triangle $$ Y_n[n] \to Y_{n + 1}[n + 1] \to X_{n + 1}[n + 1] \to Y_n[n + 1] $$ we get an exact sequence $$ \Hom(X_{n + 1}[n + 1], K') \to \Hom(Y_{n + 1}[n + 1], K') \to \Hom(Y_n[n], K') $$ As $X_{n + 1}[n + 1] = g_{n + 1!}K_{n + 1}[n + 1]$ the first group is equal to $$ \Hom(K_{n + 1}[n + 1], K'_{n + 1}) $$ which is zero by assumption (2). By induction we conclude all the maps $Y_n[n] \to K'$ are zero. Consider the defining distinguished triangle $$ \bigoplus Y_n[n] \to \bigoplus Y_n[n] \to K \to (\bigoplus Y_n[n])[1] $$ for the homotopy colimit. Arguing as above, we find that it suffices to show that $$ \Hom((\bigoplus Y_n[n])[1], K') = \prod \Hom(Y_n[n + 1], K') $$ is zero for all $n \geq 0$. To see this, arguing as above, it suffices to show that $$ \Hom(K_n[n + 1], K'_n)  = 0 $$ for all $n \geq 0$ which follows from condition (3).