hcju/stacksqa huggingface dataset:
Let $\{K_n \to K'_n\}_{n \geq 0}$ be a morphism of simplicial systems of the derived category. Consider the objects $X_n$ and Postnikov system $Y_n$ associated to $K$ of Lemma \ref{lemma-abelian-postnikov}. By (1) and Lemma \ref{lemma-derived-cartesian-shriek} we have $X_n = g_{n!}K_n$. In particular, the map $K_0 \to K'_0$ induces a morphism $X_0 \to K'$. Since $\{K_n \to K'_n\}$ is a morphism of systems, a computation (omitted) shows that the composition $$ X_1 \to X_0 \to K' $$ is zero. As $Y_0 = X_0$ and as $Y_1$ fits into a distinguished triangle $$ Y_1 \to X_1 \to Y_0 \to Y_1[1] $$ we conclude that there exists a morphism $Y_1[1] \to K'$ whose composition with $X_0 = Y_0 \to Y_1[1]$ is the morphism $X_0 \to K'$ given above. Suppose given a map $Y_n[n] \to K'$ for $n \geq 1$. From the distinguished triangle $$ X_{n + 1}[n] \to Y_n[n] \to Y_{n + 1}[n + 1] \to X_{n + 1}[n + 1] $$ we get an exact sequence $$ \Hom(Y_{n + 1}[n + 1], K') \to \Hom(Y_n[n], K') \to \Hom(X_{n + 1}[n], K') $$ As $X_{n + 1}[n] = g_{n + 1!}K_{n + 1}[n]$ the last group is equal to $$ \Hom(K_{n + 1}[n], K'_{n + 1}) $$ which is zero by assumption (2). By induction we get a system of maps $Y_n[n] \to K'$ compatible with transition maps and reducing to the given map on $Y_0$. This produces a map $$ \gamma : K = \text{hocolim} Y_n[n] \longrightarrow K' $$ This map in any case has the property that the diagram $$ \xymatrix{ X_0 \ar[rd] \ar[r] & K \ar[d]^\gamma \\ & K' } $$ is commutative. Restricting to $\mathcal{C}_0$ we deduce that the map $\gamma_0 : K_0 \to K'_0$ is the same as the first map $K_0 \to K'_0$ of the morphism of simplicial systems. Since $K$ is cartesian, this easily gives that $\{\gamma_n\}$ is the map of simplicial systems we started out with.
First, if $K = \mathcal{O}$, then this is the construction of Derived Categories, Example \ref{derived-example-key-postnikov} applied to the complex $$ \ldots \to g_{2!}\mathcal{O}_2 \to g_{1!}\mathcal{O}_1 \to g_{0!}\mathcal{O}_0 $$ in $\textit{Ab}(\mathcal{C}_{total})$ combined with the fact that this complex represents $K = \mathcal{O}$ in $D(\mathcal{C}_{total})$ by Lemma \ref{lemma-simplicial-resolution-ringed}. The general case follows from this, the fact that the exact functor $- \otimes^\mathbf{L}_\mathcal{O} K$ sends Postnikov systems to Postnikov systems, and that $- \otimes^\mathbf{L}_\mathcal{O} K$ commutes with homotopy colimits.
The proof is exactly the same as the proof of Lemma \ref{lemma-cartesian-object-derived-from-simplicial} with the following changes \begin{enumerate} \item use $g_n^* = Lg_n^*$ everywhere instead of $g_n^{-1}$, \item use $f_\varphi^* = Lf_\varphi^*$ everywhere instead of $f_\varphi^{-1}$, \item refer to Lemma \ref{lemma-simplicial-resolution-ringed} instead of Lemma \ref{lemma-simplicial-resolution-Z-site}, \item in the construction of $Y'_{m, n}$ use $\mathcal{O}_m$ instead of $\mathbf{Z}$, \item compare with the proof of Lemma \ref{lemma-modules-postnikov} rather than the proof of Lemma \ref{lemma-abelian-postnikov}. \end{enumerate} This ends the proof.
Proof of (2). In the notation of the discussion preceding the lemma, the localization functors $\mathcal{C}/U_i \to \mathcal{C}/V_{\alpha(i)}$ are continuous and cocontinuous by Sites, Section \ref{sites-section-localize} and satisfy $P$ by Sites, Remark \ref{sites-remark-localization-cartesian-cocontinuous}. It is formal to deduce $v$ is continuous and cocontinuous and has $P$. We omit the details. We also omit the proof of (1).
Say $K = \{U_i\}_{i \in I}$. Using the description of the functors $j^{-1}$ and $j_*$ given above we see that $$ j_*j^{-1}\mathcal{F} = \prod\nolimits_{i \in I} j_{i, *}(\mathcal{F}|_{\mathcal{C}/U_i}) = \prod\nolimits_{i \in I} \SheafHom(h_{U_i}^\#, \mathcal{F}) $$ The second equality by Sites, Lemma \ref{sites-lemma-hom-sheaf-hU}. Since $F(K) = \coprod h_{U_i}$ in $\textit{PSh}(\mathcal{C}$, we have $F(K)^\# = \coprod h_{U_i}^\#$ in $\Sh(\mathcal{C})$ and since $\SheafHom(-, \mathcal{F})$ turns coproducts into products (immediate from the construction in Sites, Section \ref{sites-section-glueing-sheaves}), we conclude.
Observe that if $K = \{U_i\}_{i \in I}$ then the category $\Sh(\mathcal{C}/K)$ decomposes as the product of the categories $\Sh(\mathcal{C}/U_i)$. Observe that $F(K)^\# = \coprod_{i \in I} h_{U_i}^\#$ (coproduct in sheaves). Hence $\Sh(\mathcal{C})/F(K)^\#$ is the product of the categories $\Sh(\mathcal{C})/h_{U_i}^\#$. Thus (1) and (2) follow from the corresponding statements for each $i$, see Sites, Lemmas \ref{sites-lemma-essential-image-j-shriek} and \ref{sites-lemma-compute-j-shriek-restrict}. Similarly, if $L = \{V_j\}_{j \in J}$ and $f$ is given by $\alpha : I \to J$ and $f_i : U_i \to V_{\alpha(i)}$, then we can apply Sites, Lemma \ref{sites-lemma-relocalize-explicit} to each of the re-localization morphisms $\mathcal{C}/U_i \to \mathcal{C}/V_{\alpha(i)}$ to get (3).
The first statement is the natural generalization of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open} to semi-representable objects. In fact, it follows from this lemma by the product decomposition of $\Sh(\mathcal{C}/K)$ and the description of the functor $j^{-1}$ given above. The second statement is the natural generalization of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-restrict-K-injective-to-open} and follows from it by the product decomposition of the topos. \medskip\noindent Alternative: since $j$ induces a localization of topoi by Lemma \ref{lemma-localize-compare} part (1) it also follows immediately from Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and \ref{sites-cohomology-lemma-restrict-K-injective-to-open} by enlarging the site; compare with the proof of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-on-sheaf-sets} in the case of injective sheaves.
Parts (3) and (4) hold for any augmentation of a simplicial site, see Lemma \ref{lemma-augmentation-site}. Parts (1) and (2) follow as $j_{total}$ is continuous and cocontinuous. The functor $a^{Sh}_!$ is constructed in Sites, Lemma \ref{sites-lemma-when-shriek} and the functor $a_!$ is constructed in Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint}.
By Lemma \ref{lemma-push-pull-localization} we have $$ a_{n, *}a_n^{-1}\mathcal{F} = \SheafHom'(F(K_n)^\#, \mathcal{F}) $$ where $\SheafHom'$ is as in Sites, Section \ref{sites-section-glueing-sheaves}. The boundary maps in the complex of Lemma \ref{lemma-augmentation-cech-complex} come from the simplicial structure. Thus the equality of complexes comes  from the canonical identifications $\SheafHom'(\mathcal{G}, \mathcal{F}) = \SheafHom(\mathbf{Z}_\mathcal{G}, \mathcal{F})$ for $\mathcal{G}$ in $\Sh(\mathcal{C})$.
Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma \ref{lemma-Serre-subcat-cartesian-modules}. The equivalence is a formal consequence of the results obtained so far. Use Lemmas \ref{lemma-hypercovering-descent-sheaves} and \ref{lemma-hypercovering-descent-bounded-abelian} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-equivalence-bounded}
Since $a^{-1}\mathcal{O}_\mathcal{C} = \mathcal{O}$ we have $La^* = a^* = a^{-1}$. Moreover $Ra_*$ agrees with $Ra_*$ on abelian sheaves, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-modules-abelian-unbounded}. Hence the lemma follows immediately from Lemma \ref{lemma-hypercovering-descent-bounded-abelian}.
This follows from Lemma \ref{lemma-hypercovering-descent-bounded-modules} because $R\Gamma((\mathcal{C}/K)_{total}, -) = R\Gamma(\mathcal{C}, -) \circ Ra_*$ by Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray} or by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Leray-unbounded}.
Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma \ref{lemma-Serre-subcat-cartesian-modules} (the required hypotheses hold by the discussion in Remark \ref{remark-augmentation-ringed}). The equivalence is a formal consequence of the results obtained so far. Use Lemmas \ref{lemma-hypercovering-descent-modules} and \ref{lemma-hypercovering-descent-bounded-modules} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-equivalence-bounded}.
Given a morphism of objects $V \to W$ of $\mathcal{C}$ the localization morphism $j : \mathcal{C}/V \to \mathcal{C}/W$ is a left adjoint to the base change functor $\mathcal{C}/W \to \mathcal{C}/V$. The base change functor is continuous and induces the same morphism of topoi as $j$. See Sites, Lemma \ref{sites-lemma-relocalize-given-fibre-products}. This proves (1). \medskip\noindent Part (2) holds because a morphism $V/U_n \to W/U_m$ of the category constructed in Lemma \ref{lemma-simplicial-site-site} is a morphism $V \to W \times_{U_m, f_\varphi} U_n$ over $U_n$ which is the same thing as a morphism $f : V \to W$ over the morphism $f_\varphi : U_n \to U_m$, i.e., the same thing as a morphism in the category $(\mathcal{C}/U)_{total}$ defined above. Equality of sets of coverings is immediate from the definition. \medskip\noindent We omit the proof of (3).
Recall that $\Sh(\mathcal{C}/K_n) = \prod_{i \in I_n} \Sh(\mathcal{C}/u(U_{n, i}))$ and similarly for the categories of modules. This product decomposition is also inherited by the derived categories of sheaves of modules. Moreover, this product decomposition is compatible with the morphisms in the simplicial semi-representable object $K$. See Section \ref{section-semi-representable}. Hence we can set $E_n = \prod_{i \in I_n} E_{U_{n, i}}$ (``formal'' product) in $D(\mathcal{O}_n)$. Taking (formal) products of the maps $\rho_a$ of Situation \ref{situation-locally-given} we obtain isomorphisms $E_\varphi : f_\varphi^*E_n \to E_m$. The assumption about compostions of the maps $\rho_a$ immediately implies that $(E_n, E_\varphi)$ defines a simplicial system of the derived category of modules as in Definition \ref{definition-cartesian-derived-modules}. The vanishing of negative exts assumed in the lemma implies that $\Hom(E_n[t], E_n) = 0$ for $n \geq 0$ and $t > 0$. Thus by Lemma \ref{lemma-cartesian-module-derived-from-simplicial} we obtain $E$. Uniqueness up to unique isomorphism follows from Lemmas \ref{lemma-nullity-cartesian-modules-derived} and \ref{lemma-hom-cartesian-modules-derived}.
By Hypercoverings, Lemma \ref{hypercovering-lemma-hypercovering-site} there exists a hypercovering $L$ for the site $\mathcal{D}$ such that $L_n = \{U_{n, i}\}_{i \in I_n}$ with $U_{i, n} \in \Ob(\mathcal{B})$. Set $K = u(L)$. Apply Lemma \ref{lemma-prepare-bbd-glueing} to get a cartesian object $E$ of $D(\mathcal{O})$ on the site $(\mathcal{C}/K)_{total}$ restricting to $E_{U_{n, i}}$ on $\mathcal{C}/u(U_{n, i})$ compatibly. The assumption on $t$ implies that $E \in D^+(\mathcal{O})$. By Hypercoverings, Lemma \ref{hypercovering-lemma-hypercovering-morphism-sites} we see that $K$ is a hypercovering too. By Lemma \ref{lemma-hypercovering-equivalence-bounded-modules} we find that $E = a^*F$ for some $F$ in $D^+(\mathcal{O}_\mathcal{C})$. \medskip\noindent To prove that $F$ is a solution we will use the construction of $L_0$ and $L_1$ given in the proof of Hypercoverings, Lemma \ref{hypercovering-lemma-hypercovering-site}. (This is a bit inelegant but there does not seem to be a completely straightforward way around it.) \medskip\noindent Namely, we have $I_0 = \Ob(\mathcal{B})$ and so $L_0 = \{U\}_{U \in \Ob(\mathcal{B})}$. Hence the isomorphism $a^*F \to E$ restricted to the components $\mathcal{C}/u(U)$ of $\mathcal{C}/K_0$ defines isomorphisms $\rho_U : F|_{\mathcal{C}/u(U)} \to E_U$ for $U \in \Ob(\mathcal{B})$ by our choice of $E$. \medskip\noindent To prove that $\rho_U$ satisfy the requirement of compatibility with the maps $\rho_a$ of Situation \ref{situation-locally-given} we use that $I_1$ contains the set $$ \Omega = \{(U, V, W, a, b) \mid U, V, W \in \mathcal{B}, a : U \to V, b : U \to W\} $$ and that for $i = (U, V, W, a, b)$ in $\Omega$ we have $U_{1, i} = U$. Moreover, the component maps $f_{\delta^1_0, i}$ and $f_{\delta^1_1, i}$ of the two morphisms $K_1 \to K_0$ are the morphisms $$ a : U \to V \quad\text{and}\quad b : U \to V $$ Hence the compatibility mentioned in Lemma \ref{lemma-prepare-bbd-glueing} gives that $$ \rho_a \circ \rho_V|_{\mathcal{C}/u(U)} = \rho_U \quad\text{and}\quad \rho_b \circ \rho_W|_{\mathcal{C}/u(U)} = \rho_U $$ Taking $i = (U, V, U, a, \text{id}_U) \in \Omega$ for example, we find that we have the desired compatibility. The uniqueness of $F$ follows from the uniqueness of $E$ in the previous lemma (small detail omitted).
The proof is {\bf exactly} the same as the proof of Lemma \ref{lemma-bbd-glueing}. The only change is that $E$ is an object of $D_{\mathcal{A}_{total}}(\mathcal{O})$ and hence we use Lemma \ref{lemma-hypercovering-equivalence-modules} to obtain $F$ with $E = a^*F$ instead of Lemma \ref{lemma-hypercovering-equivalence-bounded-modules}.
Write $\Sh(X) = \Sh(X_{Zar})$. Observe that both $(\textit{LC}_{qc}/U)_{total}$ and $U_{Zar}$ fall into case A of Situation \ref{situation-simplicial-site}. This is immediate from the construction of $U_{Zar}$ in Section \ref{section-simplicial-top} and it follows from Lemma \ref{lemma-sr-when-fibre-products} for $(\textit{LC}_{qc}/U)_{total}$. Next, consider the functors $U_{n, Zar} \to \textit{LC}_{qc}/U_n$, $U \mapsto U/U_n$ and $X_{Zar} \to \textit{LC}_{qc}/X$, $U \mapsto U/X$. We have seen that these define morphisms of sites in Cohomology on Sites, Section \ref{sites-cohomology-section-cohomology-LC}. Thus we obtain a morphism of simplicial sites compatible with augmentations as in Remark \ref{remark-morphism-augmentation-simplicial-sites} and we may apply Lemma \ref{lemma-morphism-augmentation-simplicial-sites} to conclude.
We will prove the statement for sheaves of sets. It will be an almost formal consequence of results already established. Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects}. By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-describe-pullback-pi} the functor $(h_{-1})^{-1}$ is fully faithful with quasi-inverse $h_{-1, *}$. The same holds true for the components $h_n$ of $h$. By the description of the functors $h^{-1}$ and $h_*$ of Lemma \ref{lemma-morphism-simplicial-sites} we conclude that $h^{-1}$ is fully faithful with quasi-inverse $h_*$. Observe that $U$ is a hypercovering of $X$ in $\textit{LC}_{qc}$ (as defined in Section \ref{section-hypercovering}) by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-proper-surjective-is-qc-covering}. By Lemma \ref{lemma-hypercovering-X-simple-descent-sheaves} we see that $a_{qc}^{-1}$ is fully faithful with quasi-inverse $a_{qc, *}$ and with essential image the cartesian sheaves on $(\textit{LC}_{qc}/U)_{total}$. A formal argument (chasing around the diagram) now shows that $a^{-1}$ is fully faithful. \medskip\noindent Finally, suppose that $\mathcal{G}$ is a cartesian sheaf on $U_{Zar}$. Then $h^{-1}\mathcal{G}$ is a cartesian sheaf on $\textit{LC}_{qc}/U$. Hence $h^{-1}\mathcal{G} = a_{qc}^{-1}\mathcal{H}$ for some sheaf $\mathcal{H}$ on $\textit{LC}_{qc}/X$. We compute \begin{align*} (h_{-1})^{-1}(a_*\mathcal{G}) & = (h_{-1})^{-1} \text{Eq}( \xymatrix{ a_{0, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ (h_{-1})^{-1}a_{0, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & (h_{-1})^{-1}a_{1, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{qc, 0, *}h_0^{-1}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{qc, 1, *}h_1^{-1}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{qc, 0, *}a_{qc, 0}^{-1}\mathcal{H} \ar@<1ex>[r] \ar@<-1ex>[r] & a_{qc, 1, *}a_{qc, 1}^{-1}\mathcal{H} } ) \\ & = a_{qc, *}a_{qc}^{-1}\mathcal{H} \\ & = \mathcal{H} \end{align*} Here the first equality follows from Lemma \ref{lemma-augmentation}, the second equality follows as $(h_{-1})^{-1}$ is an exact functor, the third equality follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-push-pull-LC} (here we use that $a_0 : U_0 \to X$ and $a_1: U_1 \to X$ are proper), the fourth follows from $a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$, the fifth from Lemma \ref{lemma-augmentation-site}, and the sixth we've seen above. Since $a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$ we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}$ which ends the proof by fully faithfulness of $h^{-1}$.
Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects}. Observe that $Rh_{n, *}h_n^{-1}$ is the identity functor on $D^+(U_n)$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomological-descent-LC}. Hence $Rh_*h^{-1}$ is the identity functor on $D^+(U_{Zar})$ by Lemma \ref{lemma-direct-image-morphism-simplicial-sites}. We have \begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{qc, *}a_{qc}^{-1}(h_{-1})^{-1}K \\ & = Rh_{-1, *}(h_{-1})^{-1}K \\ & = K \end{align*} The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma \ref{lemma-compare-simplicial-objects}, the third equality by Lemma \ref{lemma-hypercovering-X-simple-descent-bounded-abelian} ($U$ is a hypercovering of $X$ in $\textit{LC}_{qc}$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-proper-surjective-is-qc-covering}), and the last equality by the already used Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomological-descent-LC}.
Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma \ref{lemma-Serre-subcat-cartesian-modules}. The equivalence is a formal consequence of the results obtained so far. Use Lemmas \ref{lemma-descent-sheaves-for-proper-hypercovering} and \ref{lemma-cohomological-descent-for-proper-hypercovering} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-equivalence-bounded}.
The statement of the displayed equality makes sense because $d^1_1 \circ d^2_2 = d^1_1 \circ d^2_1$, $d^1_1 \circ d^2_0 = d^1_0 \circ d^2_2$, and $d^1_0 \circ d^2_0 = d^1_0 \circ d^2_1$ as morphisms $X_2 \to X_0$, see Simplicial, Remark \ref{simplicial-remark-relations} hence we can picture these maps as follows $$ \xymatrix{ & X_2 \times_{d^1_1 \circ d^2_0, X_0} V \ar[r]_-{(d^2_0)^*\varphi} & X_2 \times_{d^1_0 \circ d^2_0, X_0} V \ar@{=}[rd] & \\ X_2 \times_{d^1_0 \circ d^2_2, X_0} V \ar@{=}[ru] & & & X_2 \times_{d^1_0 \circ d^2_1, X_0} V \\ & X_2 \times_{d^1_1 \circ d^2_2, X_0} V \ar[lu]^{(d^2_2)^*\varphi} \ar@{=}[r] & X_2 \times_{d^1_1 \circ d^2_1, X_0} V \ar[ru]_{(d^2_1)^*\varphi} } $$ and the condition signifies the diagram is commutative. It is clear that given a simplicial scheme $Y$ cartesian over $X$ we can set $V = Y_0$ and $\varphi$ equal to the composition $$ V \times_{X_0, d^1_1} X_1 = Y_0 \times_{X_0, d^1_1} X_1 = Y_1 = X_1 \times_{X_0, d^1_0} Y_0 = X_1 \times_{X_0, d^1_0} V $$ of identifications given by the cartesian structure. 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 $(V, \varphi)$ as in the lemma we set $Y_n = X_n \times_{X(\tau^n_n), X_0} V$. Then given $\beta : [n] \to [m]$ we define $V(\beta) : Y_m \to Y_n$ as the pullback by $X(\tau^m_{\beta(n)m})$ of the map $\varphi$ postcomposed by the projection $X_m \times_{X(\beta), X_n} Y_n \to Y_n$. This makes sense because $$ X_m \times_{X(\tau^m_{\beta(n)m}), X_1} X_1 \times_{d^1_1, X_0} V = X_m \times_{X(\tau^m_m), X_0} V = Y_m $$ and $$ X_m \times_{X(\tau^m_{\beta(n)m}), X_1} X_1 \times_{d^1_0, X_0} V = X_m \times_{X(\tau^m_{\beta(n)}), X_0} V = X_m \times_{X(\beta), X_n} Y_n. $$ We omit the verification that the commutativity of the displayed diagram above implies the maps compose correctly. We also omit the verification that the two functors are quasi-inverse to each other.
Recall that $\mathcal{F}$ is cartesian with $\mathcal{F}_n$ quasi-coherent, see Lemma \ref{lemma-quasi-coherent-sheaf}. By Lemma \ref{lemma-describe-functoriality} we see that $(f^*\mathcal{F})_n = f_n^*\mathcal{F}_n$ (some details omitted). Hence $(f^*\mathcal{F})_n$ is quasi-coherent. The same fact and the cartesian property for $\mathcal{F}$ imply the cartesian property for $f^*\mathcal{F}$. Thus $\mathcal{F}$ is quasi-coherent by Lemma \ref{lemma-quasi-coherent-sheaf} again.
If $\mathcal{F} = f_* \mathcal{G}$, then $\mathcal{F}_n = f_{n , *}\mathcal{G}_n$ by Lemma \ref{lemma-describe-functoriality}. The maps $\mathcal{F}(\varphi)$ are defined using the base change maps, see Cohomology, Section \ref{cohomology-section-base-change-map}. The sheaves $\mathcal{F}_n$ are quasi-coherent by Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} and the fact that $\mathcal{G}_n$ is quasi-coherent by Lemma \ref{lemma-quasi-coherent-sheaf}. The base change maps along the degeneracies $d^n_j$ are isomorphisms by Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology} and the fact that $\mathcal{G}$ is cartesian by Lemma \ref{lemma-quasi-coherent-sheaf}. Hence $\mathcal{F}$ is cartesian by Lemma \ref{lemma-check-cartesian-module}. Thus $\mathcal{F}$ is quasi-coherent by Lemma \ref{lemma-quasi-coherent-sheaf}.
Let $\sigma : S \to X$ be a section of $f$. Let $(\mathcal{F}, \alpha)$ be a pair as in Lemma \ref{lemma-characterize-cartesian-modules}. Set $\mathcal{G} = \sigma^*\mathcal{F}$. Consider the diagram $$ \xymatrix{ X \ar[r]_-{(\sigma \circ f, 1)} \ar[d]_f & X \times_S X \ar[d]^{\text{pr}_0} \ar[r]_-{\text{pr}_1} & X \\ S \ar[r]^\sigma & X } $$ Note that $\text{pr}_0 = d^1_1$ and $\text{pr}_1 = d^1_0$. Hence we see that $(\sigma \circ f, 1)^*\alpha$ defines an isomorphism $$ f^*\mathcal{G} = (\sigma \circ f, 1)^*\text{pr}_0^*\mathcal{F} \longrightarrow (\sigma \circ f, 1)^*\text{pr}_1^*\mathcal{F} = \mathcal{F} $$ We omit the verification that this isomorphism is compatible with $\alpha$ and the canonical isomorphism $\text{pr}_0^*f^*\mathcal{G} \to \text{pr}_1^*f^*\mathcal{G}$.
We only have to verify that the rules prescribed in (1), (2), (3), (4), (5) define a $2$-truncated simplicial scheme $U'$ over $S$, since then (6) allows us to set $X = \text{cosk}_2 U'$, see Simplicial, Lemma \ref{simplicial-lemma-existence-cosk}. Using the functor of points approach, all we have to verify is that if $(\text{Ob}, \text{Arrows}, s, t, c, e, i)$ is a groupoid, then $$ \xymatrix{ \text{Arrows} \times_{s, \text{Ob}, t} \text{Arrows} \ar@<8ex>[d]^{\text{pr}_0} \ar@<0ex>[d]_c \ar@<-8ex>[d]_{\text{pr}_1} \\ \text{Arrows} \ar@<4ex>[d]^t \ar@<-4ex>[d]_s \ar@<4ex>[u]^{1, e} \ar@<-4ex>[u]_{e, 1} \\ \text{Ob} \ar@<0ex>[u]_e } $$ is a $2$-truncated simplicial set. We omit the details. \medskip\noindent Finally, the description of $X_n$ for $n > 2$ follows by induction from the description of $X_0$, $X_1$, $X_2$, and Simplicial, Remark \ref{simplicial-remark-inductive-coskeleton} and Lemma \ref{simplicial-lemma-work-out}. Alternately, one shows that $\text{cosk}_2$ applied to the $2$-truncated simplicial set displayed above gives a simplicial set whose $n$th term equals $\text{Arrows} \times_{s, \text{Ob}, t} \ldots \times_{s, \text{Ob}, t} \text{Arrows}$ with $n$ factors and degeneracy maps as given in the lemma. Some details omitted.
For arbitrary $n$ we define $(R/U)_\bullet \to X_n$ by the rule $$ (r_0, \ldots, r_n) \longrightarrow (r_0 \circ r_1^{-1}, \ldots, r_{n - 1} \circ r_n^{-1}) $$ Compatibility with degeneracy maps is clear from the description of the degeneracies in Lemma \ref{lemma-groupoid-simplicial}. We omit the verification that the maps respect the morphisms $s^n_j$. Groupoids, Lemma \ref{groupoids-lemma-diagram-pull} (with the roles of $s$ and $t$ reversed) shows that the two right squares are cartesian. In exactly the same manner one shows all the other squares are cartesian too. Hence the morphism is cartesian.