hcju/stacksqa huggingface dataset:
Note that $j$ is a monomorphism. Namely the composition $Y_1 \to Y_0 \times_S Y_0 \to Y_0 \times_S X$ is an isomorphism as $\pi$ is cartesian. \medskip\noindent Consider the morphism $$ (d^2_2, d^2_0) : Y_2 \to Y_1 \times_{d^1_0, Y_0, d^1_1} Y_1. $$ This works because $d_0 \circ d_2 = d_1 \circ d_0$, see Simplicial, Remark \ref{simplicial-remark-relations}. Also, it is a morphism over $(X/S)_2$. It is an isomorphism because $Y \to (X/S)_\bullet$ is cartesian. Note for example that the right hand side is isomorphic to $Y_0 \times_{\pi_0, X, \text{pr}_1} (X \times_S X \times_S X) = X \times_S Y_0 \times_S X$ because $\pi$ is cartesian. Details omitted. \medskip\noindent As in Groupoids, Definition \ref{groupoids-definition-equivalence-relation} we denote $t = \text{pr}_0 \circ j = d^1_1$ and $s = \text{pr}_1 \circ j = d^1_0$. The isomorphism above, combined with the morphism $d^2_1 : Y_2 \to Y_1$ give us a composition morphism $$ c : Y_1 \times_{s, Y_0, t} Y_1 \longrightarrow Y_1 $$ over $Y_0 \times_S Y_0$. This immediately implies that for any scheme $T/S$ the relation $Y_1(T) \subset Y_0(T) \times Y_0(T)$ is transitive. \medskip\noindent Reflexivity follows from the fact that the restriction of the morphism $j$ to the diagonal $\Delta : X \to X \times_S X$ is an isomorphism (again use the cartesian property of $\pi$). \medskip\noindent To see symmetry we consider the morphism $$ (d^2_2, d^2_1) : Y_2 \to Y_1 \times_{d^1_1, Y_0, d^1_1} Y_1. $$ This works because $d_1 \circ d_2 = d_1 \circ d_1$, see Simplicial, Remark \ref{simplicial-remark-relations}. It is an isomorphism because $Y \to (X/S)_\bullet$ is cartesian. Note for example that the right hand side is isomorphic to $Y_0 \times_{\pi_0, X, \text{pr}_0} (X \times_S X \times_S X) = Y_0 \times_S X \times_S X$ because $\pi$ is cartesian. Details omitted. \medskip\noindent Let $T/S$ be a scheme. Let $a \sim b$ for $a, b \in Y_0(T)$ be synonymous with $(a, b) \in Y_1(T)$. The isomorphism $(d^2_2, d^2_1)$ above implies that if $a \sim b$ and $a \sim c$, then $b \sim c$. Combined with reflexivity this shows that $\sim$ is an equivalence relation.
Let $Y \to (X/S)_\bullet$ be the cartesian morphism of simplicial schemes corresponding to $(V, \varphi)$ by Lemma \ref{lemma-cartesian-equivalent-descent-datum}. Observe that $Y_0 = V$. Write $V = \coprod_{i \in I} \Spec(A_i)$ with each $A_i$ local Artinian. Moreover, let $v_i \in V$ be the unique closed point of $\Spec(A_i)$ for all $i \in I$. Write $i \sim j$ if and only if $v_i \in T_{v_j}$ with notation as in Lemma \ref{lemma-equivalence-classes-points} above. By Lemmas \ref{lemma-equivalence-classes-points} and \ref{lemma-quasi-compact} this is an equivalence relation with finite equivalence classes. Let $\overline{I} = I/\sim$. Then we can write $V = \coprod_{\overline{i} \in \overline{I}} V_{\overline{i}}$ with $V_{\overline{i}} = \coprod_{i \in \overline{i}} \Spec(A_i)$. By construction we see that $\varphi : V \times_S X \to X \times_S V$ maps the open and closed subspaces $V_{\overline{i}} \times_S X$ into the open and closed subspaces $X \times_S V_{\overline{i}}$. In other words, we get descent data $(V_{\overline{i}}, \varphi_{\overline{i}})$, and $(V, \varphi)$ is the coproduct of them in the category of descent data. Since each of the $V_{\overline{i}}$ is a finite union of spectra of Artinian local rings the morphism $V_{\overline{i}} \to X$ is affine, see Morphisms, Lemma \ref{morphisms-lemma-Artinian-affine}. Since $\{X \to S\}$ is an fpqc covering we see that all the descent data $(V_{\overline{i}}, \varphi_{\overline{i}})$ are effective by Descent, Lemma \ref{descent-lemma-affine}.
The notation $(\textit{Spaces}/U)_{fppf, total}$ indicates that we are using the construction of Section \ref{section-hypercovering} for the site $(\textit{Spaces}/S)_{fppf}$ and the simplicial object $U$ of this site\footnote{We could also use the \'etale topology and this would be denoted $(\textit{Spaces}/U)_{\etale, total}$.}. We will use the sites $X_{spaces, \etale}$ and $U_{spaces, \etale}$ for the topoi on the right hand side; this is permissible see discussion in Section \ref{section-simplicial-algebraic-spaces}. \medskip\noindent Observe that both $(\textit{Spaces}/U)_{fppf, total}$ and $U_{spaces, \etale}$ fall into case A of Situation \ref{situation-simplicial-site}. This is immediate from the construction of $U_\etale$ in Section \ref{section-simplicial-algebraic-spaces} and it follows from Lemma \ref{lemma-sr-when-fibre-products} for $(\textit{Spaces}/U)_{fppf, total}$. Next, consider the functors $U_{n, spaces, \etale} \to (\textit{Spaces}/U_n)_{fppf}$, $U \mapsto U/U_n$ and $X_{spaces, \etale} \to (\textit{Spaces}/X)_{fppf}$, $U \mapsto U/X$. We have seen that these define morphisms of sites in More on Cohomology of Spaces, Section \ref{spaces-more-cohomology-section-fppf-etale} where these were denoted $a_{U_n} = \epsilon_{U_n} \circ \pi_{u_n}$ and $a_X = \epsilon_X \circ \pi_X$. 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-fppf-etale}. In the proof of this lemma we have seen that $h_{-1}$ is the morphism $a_X$ of More on Cohomology of Spaces, Section \ref{spaces-more-cohomology-section-fppf-etale}. Thus it follows from More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-comparison-fppf-etale} that $(h_{-1})^{-1}$ 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 $(\textit{Spaces}/S)_{fppf}$ as defined in Section \ref{section-hypercovering}. By Lemma \ref{lemma-hypercovering-X-simple-descent-sheaves} we see that $a_{fppf}^{-1}$ is fully faithful with quasi-inverse $a_{fppf, *}$ and with essential image the cartesian sheaves on $(\textit{Spaces}/U)_{fppf, total}$. 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 $U_\etale$. Then $h^{-1}\mathcal{G}$ is a cartesian sheaf on $(\textit{Spaces}/U)_{fppf, total}$. Hence $h^{-1}\mathcal{G} = a_{fppf}^{-1}\mathcal{H}$ for some sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{fppf}$. In particular we find that $h_0^{-1}\mathcal{G}_0 = (a_{0, big, fppf})^{-1}\mathcal{H}$. Recalling that $h_0 = a_{U_0}$ and that $U_0 \to X$ is flat, locally of finite presentation, and surjective, we find from More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-descent-sheaf-fppf-etale} that there exists a sheaf $\mathcal{F}$ on $X_\etale$ and isomorphism $\mathcal{H} = (h_{-1})^{-1}\mathcal{F}$. Since $a_{fppf}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$ we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}\mathcal{F}$. By fully faithfulness of $h^{-1}$ we conclude that $a^{-1}\mathcal{F} \cong \mathcal{G}$. \medskip\noindent Fix an isomorphism $\theta : a^{-1}\mathcal{F} \to \mathcal{G}$. To finish the proof we have to show $\mathcal{G} = a^{-1}a_*\mathcal{G}$ (in order to show that the quasi-inverse is given by $a_*$; everything else has been proven above). Because $a^{-1}$ is fully faithful we have $\text{id} \cong a_*a^{-1}$ by Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}. Thus $\mathcal{F} \cong a_*a^{-1}\mathcal{F}$ and $a_*\theta : a_*a^{-1}\mathcal{F} \to a_*\mathcal{G}$ combine to an isomorphism $\mathcal{F} \to a_*\mathcal{G}$. Pulling back by $a$ and precomposing by $\theta^{-1}$ we find the desired isomorphism.
Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects-fppf-etale}. Observe that $Rh_{n, *}h_n^{-1}$ is the identity functor on $D^+(U_{n, \etale})$ by More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf}. Hence $Rh_*h^{-1}$ is the identity functor on $D^+(U_\etale)$ 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_{fppf, *}a_{fppf}^{-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} as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$, and the last equality by the already used More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf}.
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-fppf-hypercovering} and \ref{lemma-cohomological-descent-for-fppf-hypercovering} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-equivalence-bounded}.
For the underlying diagram of topoi we refer to the discussion in the proof of Lemma \ref{lemma-compare-simplicial-objects-fppf-etale}. The sheaf $\mathcal{O}_U$ is the structure sheaf of the simplicial algebraic space $U$ as defined in Section \ref{section-simplicial-algebraic-spaces}. The sheaf $\mathcal{O}_X$ is the usual structure sheaf of the algebraic space $X$. The sheaves of rings $\mathcal{O}_{big, total}$ and $\mathcal{O}_{big}$ come from the structure sheaf on $(\textit{Spaces}/S)_{fppf}$ in the manner explained in Section \ref{section-hypercovering-modules} which also constructs $a_{fppf}$ as a morphism of ringed topoi. The component morphisms $h_n = a_{U_n}$ and $h_{-1} = a_X$ are morphisms of ringed topoi by More on Cohomology of Spaces, Section \ref{spaces-more-cohomology-section-fppf-etale-modules}. Finally, since the continuous functor $u : U_{spaces, \etale} \to (\textit{Spaces}/U)_{fppf, total}$ used to define $h$\footnote{This happened in the proof of Lemma \ref{lemma-compare-simplicial-objects-fppf-etale} via an application of Lemma \ref{lemma-morphism-augmentation-simplicial-sites}.} is given by $V/U_n \mapsto V/U_n$ we see that $h_*\mathcal{O}_{big, total} = \mathcal{O}_U$ which is how we endow $h$ with the structure of a morphism of ringed simplicial sites as in Remark \ref{remark-morphism-simplicial-sites-modules}. Then we obtain $h$ as a morphism of ringed topoi by Lemma \ref{lemma-morphism-simplicial-sites-modules}. Please observe that the morphisms $h_n$ indeed agree with the morphisms $a_{U_n}$ described above. We omit the verification that the diagram is commutative (as a diagram of ringed topoi -- we already know it is commutative as a diagram of topoi).
Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects-fppf-etale-modules}. In the proof of this lemma we have seen that $h_{-1}$ is the morphism $a_X$ of More on Cohomology of Spaces, Section \ref{spaces-more-cohomology-section-fppf-etale-modules}. Thus it follows from More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-review-quasi-coherent} that $$ (h_{-1})^* : \QCoh(\mathcal{O}_X) \longrightarrow \QCoh(\mathcal{O}_{big}) $$ is an equivalence with quasi-inverse $h_{-1, *}$. The same holds true for the components $h_n$ of $h$. Recall that $\QCoh(\mathcal{O}_U)$ and $\QCoh(\mathcal{O}_{big, total})$ consist of cartesian modules whose components are quasi-coherent, see Lemma \ref{lemma-quasi-coherent-sheaf}. Since the functors $h^*$ and $h_*$ of Lemma \ref{lemma-morphism-simplicial-sites-modules} agree with the functors $h_n^*$ and $h_{n, *}$ on components we conclude that $$ h^* : \QCoh(\mathcal{O}_U) \longrightarrow  \QCoh(\mathcal{O}_{big, total}) $$ is an equivalence with quasi-inverse $h_*$. Observe that $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$ as defined in Section \ref{section-hypercovering}. By Lemma \ref{lemma-hypercovering-X-simple-descent-modules} we see that $a_{fppf}^*$ is fully faithful with quasi-inverse $a_{fppf, *}$ and with essential image the cartesian sheaves of $\mathcal{O}_{fppf, total}$-modules. Thus, by the description of $\QCoh(\mathcal{O}_{big})$ and $\QCoh(\mathcal{O}_{big, total})$ of Lemma \ref{lemma-quasi-coherent-sheaf}, we get an equivalence $$ a_{fppf}^* : \QCoh(\mathcal{O}_{big}) \longrightarrow \QCoh(\mathcal{O}_{big, total}) $$ with quasi-inverse given by $a_{fppf, *}$. A formal argument (chasing around the diagram) now shows that $a^*$ is fully faithful on $\QCoh(\mathcal{O}_X)$ and has image contained in $\QCoh(\mathcal{O}_U)$. \medskip\noindent Finally, suppose that $\mathcal{G}$ is in $\QCoh(\mathcal{O}_U)$. Then $h^*\mathcal{G}$ is in $\QCoh(\mathcal{O}_{big, total})$. Hence $h^*\mathcal{G} = a_{fppf}^*\mathcal{H}$ with $\mathcal{H} = a_{fppf, *}h^*\mathcal{G}$ in $\QCoh(\mathcal{O}_{big})$ (see above). In turn we see that $\mathcal{H} = (h_{-1})^*\mathcal{F}$ with $\mathcal{F} = h_{-1, *}\mathcal{H}$ in $\QCoh(\mathcal{O}_X)$. Going around the diagram we deduce that $h^*\mathcal{G} \cong h^*a^*\mathcal{F}$. By fully faithfulness of $h^*$ we conclude that $a^*\mathcal{F} \cong \mathcal{G}$. Since $\mathcal{F} = h_{-1, *}a_{fppf, *}h^*\mathcal{G} = a_*h_*h^*\mathcal{G} = a_*\mathcal{G}$ we also obtain the statement that the quasi-inverse is given by $a_*$.
Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects-fppf-etale}. Let $\mathcal{F}_n = a_n^*\mathcal{F}$ be the $n$th component of $a^*\mathcal{F}$. This is a quasi-coherent $\mathcal{O}_{U_n}$-module. Then $\mathcal{F}_n = Rh_{n, *}h_n^*\mathcal{F}_n$ by More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules}. Hence $a^*\mathcal{F} = Rh_*h^*a^*\mathcal{F}$ by Lemma \ref{lemma-direct-image-morphism-simplicial-sites-modules}. We have \begin{align*} Ra_*(a^*\mathcal{F}) & = Ra_*Rh_*h^*a^*\mathcal{F} \\ & = Rh_{-1, *}Ra_{fppf, *}a_{fppf}^*(h_{-1})^*\mathcal{F} \\ & = Rh_{-1, *}(h_{-1})^*\mathcal{F} \\ & = \mathcal{F} \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-modules} as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$ and $La_{fppf}^* = a_{fppf}^*$ as $a_{fppf}$ is flat (namely $a_{fppf}^{-1}\mathcal{O}_{big} = \mathcal{O}_{big, total}$, see Remark \ref{remark-augmentation-ringed}), and the last equality by the already used More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules}.
First observe that the maps $a_n : U_n \to X$ and $d^n_i : U_n \to U_{n - 1}$ are flat, locally of finite presentation, and surjective by Hypercoverings, Remark \ref{hypercovering-remark-P-covering}. \medskip\noindent Recall that an $\mathcal{O}_U$-module $\mathcal{F}$ is quasi-coherent if and only if it is cartesian and $\mathcal{F}_n$ is quasi-coherent for all $n$. See Lemma \ref{lemma-quasi-coherent-sheaf}. By Lemma \ref{lemma-Serre-subcat-cartesian-modules} (and flatness of the maps $d^n_i : U_n \to U_{n - 1}$ shown above) the cartesian modules for a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_U)$. On the other hand $\QCoh(\mathcal{O}_{U_n}) \subset \textit{Mod}(\mathcal{O}_{U_n})$ is a weak Serre subcategory for each $n$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-properties-quasi-coherent}). Combined we see that $\QCoh(\mathcal{O}_U) \subset \textit{Mod}(\mathcal{O}_U)$ is a weak Serre subcategory. \medskip\noindent To finish the proof we check the conditions (1) -- (5) of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-equivalence-unbounded-one} one by one. \medskip\noindent Ad (1). This holds since $a_n$ flat (seen above) implies $a$ is flat by Lemma \ref{lemma-flat-augmentation-modules}. \medskip\noindent Ad (2). This is the content of Lemma \ref{lemma-descent-qcoh-for-fppf-hypercovering}. \medskip\noindent Ad (3). This is the content of Lemma \ref{lemma-cohomological-descent-qcoh-for-fppf-hypercovering}. \medskip\noindent Ad (4). Recall that we can use either the site $U_\etale$ or $U_{spaces, \etale}$ to define the small \'etale topos $\Sh(U_\etale)$, see Section \ref{section-simplicial-algebraic-spaces}. The assumption of Cohomology on Sites, Situation \ref{sites-cohomology-situation-olsson-laszlo} holds for the triple $(U_{spaces, \etale}, \mathcal{O}_U, \QCoh(\mathcal{O}_U))$ and by the same reasoning for the triple $(U_\etale, \mathcal{O}_U, \QCoh(\mathcal{O}_U))$. Namely, take $$ \mathcal{B} \subset \Ob(U_\etale) \subset \Ob(U_{spaces, \etale}) $$ to be the set of affine objects. For $V/U_n \in \mathcal{B}$ take $d_{V/U_n} = 0$ and take $\text{Cov}_{V/U_n}$ to be the set of \'etale coverings $\{V_i \to V\}$ with $V_i$ affine. Then we get the desired vanishing because for $\mathcal{F} \in \QCoh(\mathcal{O}_U)$ and any $V/U_n \in \mathcal{B}$ we have $$ H^p(V/U_n, \mathcal{F}) = H^p(V, \mathcal{F}_n) $$ by Lemma \ref{lemma-sanity-check-modules}. Here on the right hand side we have the cohomology of the quasi-coherent sheaf $\mathcal{F}_n$ on $U_n$ over the affine obect $V$ of $U_{n, \etale}$. This vanishes for $p > 0$ by the discussion in Cohomology of Spaces, Section \ref{spaces-cohomology-section-higher-direct-image} and Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. \medskip\noindent Ad (5). Follows by taking $\mathcal{B} \subset \Ob(X_{spaces, \etale})$ the set of affine objects and the references given above.
Write $K_n = La_n^*K$ and $E_n = La_n^*E$. Then these are the simplicial systems of the derived category of modules (Definition \ref{definition-cartesian-derived-modules}) associated to $La^*K$ and $La^*E$ (Lemma \ref{lemma-cartesian-objects-derived-modules}) where $a : U_\etale \to X_\etale$ is as in Section \ref{section-simplicial-algebraic-spaces}. Let us prove (2) first. By Lemma \ref{lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering} we have $$ \Hom_{\mathcal{O}_X}(K, E) = \Hom_{\mathcal{O}_U}(La^*K, La^*E) $$ Thus the sequence looks like this: $$ 0 \to \Hom_{\mathcal{O}_U}(La^*K, La^*E) \to \Hom_{\mathcal{O}_{U_0}}(K_0, E_0) \to \Hom_{\mathcal{O}_{U_1}}(K_1, E_1) $$ The first arrow  is injective by Lemma \ref{lemma-nullity-cartesian-modules-derived}. The image of this arrow is the kernel of the second by Lemma \ref{lemma-hom-cartesian-modules-derived}. This finishes the proof of (2). Part (1) follows by applying part (2) with $K[i]$ and $E$ for $i > 0$.
We claim that the objects $K_n$ form the members of a  simplicial system of the derived category of modules (Definition \ref{definition-cartesian-derived-modules}) of the ringed simplicial site $U_\etale$ of Section \ref{section-simplicial-algebraic-spaces}. The construction 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 $\tau^n_{ij} : [1] \to [n]$, $0 \mapsto i$, $1 \mapsto j$. Given $\varphi : [n] \to [m]$ we define $K_\varphi : L(f_\varphi)^*K_n \to K_m$ using $$ \xymatrix{ L(f_\varphi)^*K_n \ar@{=}[r] & L(f_\varphi)^* L(f_{\tau^n_n})^*K_0 \ar@{=}[r] & L(f_{\tau^m_{\varphi(n)}})^*K_0 \ar@{=}[r] & L(f_{\tau^m_{\varphi(n)m}})^* L(f_{\delta^1_1})^*K_0 \ar[d]_{L(f_{\tau^m_{\varphi(n)m}})^*\alpha} \\ & K_m \ar@{=}[r] & L(f_{\tau^m_m})^*K_0 \ar@{=}[r] & L(f_{\tau^m_{\varphi(n)m}})^* L(f_{\delta^1_0})^*K_0 } $$ We omit the verification that the cocycle condition implies the maps compose correctly (in their respective derived categories) and hence give rise to a simplicial systems of the derived category of modules\footnote{This verification is the same as that done in the proof of Lemma \ref{lemma-characterize-cartesian} as well as in the chapter on descent referenced above. We should probably write this as a general lemma about fibred and cofibred categories over $\Delta$.}. Once this is verified, we obtain an object $K' \in D_\QCoh(\mathcal{O}_{U_\etale})$ such that $(K_n, K_\varphi)$ is the system deduced from $K'$, see Lemma \ref{lemma-cartesian-module-derived-from-simplicial}. Finally, we apply Lemma \ref{lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering} to see that $K' = La^*K$ for some $K \in D_\QCoh(\mathcal{O}_X)$ as desired.
The notation $(\textit{Spaces}/U)_{ph, total}$ indicates that we are using the construction of Section \ref{section-hypercovering} for the site $(\textit{Spaces}/S)_{ph}$ and the simplicial object $U$ of this site\footnote{To distinguish from $(\textit{Spaces}/U)_{fppf, total}$ defined using the fppf topology in Section \ref{section-fppf-hypercovering}.}. We will use the sites $X_{spaces, \etale}$ and $U_{spaces, \etale}$ for the topoi on the right hand side; this is permissible see discussion in Section \ref{section-simplicial-algebraic-spaces}. \medskip\noindent Observe that both $(\textit{Spaces}/U)_{ph, total}$ and $U_{spaces, \etale}$ fall into case A of Situation \ref{situation-simplicial-site}. This is immediate from the construction of $U_\etale$ in Section \ref{section-simplicial-algebraic-spaces} and it follows from Lemma \ref{lemma-sr-when-fibre-products} for $(\textit{Spaces}/U)_{ph, total}$. Next, consider the functors $U_{n, spaces, \etale} \to (\textit{Spaces}/U_n)_{ph}$, $U \mapsto U/U_n$ and $X_{spaces, \etale} \to (\textit{Spaces}/X)_{ph}$, $U \mapsto U/X$. We have seen that these define morphisms of sites in More on Cohomology of Spaces, Section \ref{spaces-more-cohomology-section-ph-etale} where these were denoted $a_{U_n} = \epsilon_{U_n} \circ \pi_{u_n}$ and $a_X = \epsilon_X \circ \pi_X$. 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.