hcju/stacksqa huggingface dataset:
Using Lemma \ref{lemma-omega-locally-quasi-coherent} it follows that $\mathcal{H} = \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}$ also satisfies assumptions (1) and (2) of Proposition \ref{proposition-compute-cohomology}. Write $M(n)_e = \Gamma((X, T(n)_e, \delta(n)), \mathcal{F})$ so that $M(n) = \lim_e M(n)_e$. Then \begin{align*} \lim_e \Gamma((X, T(n)_e, \delta(n)), \mathcal{H}) & = \lim_e M(n)_e \otimes_{D(n)_e} \Omega_{D(n)}/p^e\Omega_{D(n)} \\ & = \lim_e M(n)_e \otimes_{D(n)} \Omega_{D(n)} \end{align*} By Lemma \ref{lemma-vanishing} the cosimplicial modules $$ M(0)_e \otimes_{D(0)} \Omega^i_{D(0)} \to M(1)_e \otimes_{D(1)} \Omega^i_{D(1)} \to M(2)_e \otimes_{D(2)} \Omega^i_{D(2)} \to \ldots $$ are homotopic to zero. Because the transition maps $M(n)_{e + 1} \to M(n)_e$ are surjective, we see that the inverse limit of the associated complexes are acyclic\footnote{Actually, they are even homotopic to zero as the homotopies fit together, but we don't need this. The reason for this roundabout argument is that the limit $\lim_e M(n)_e \otimes_{D(n)} \Omega^i_{D(n)}$ isn't the $p$-adic completion of $M(n) \otimes_{D(n)} \Omega^i_{D(n)}$ as with the assumptions of the lemma we don't know that $M(n)_e = M(n)_{e + 1}/p^eM(n)_{e + 1}$. If $\mathcal{F}$ is a crystal then this does hold.}. Hence the vanishing of cohomology of $\mathcal{H}$ by Proposition \ref{proposition-compute-cohomology}.
Choose $a : D \to D'$ and $b : D' \to D$ as in Lemma \ref{lemma-crystals-on-affine-smooth}. Note that the base change $M = M' \otimes_{D', b} D$ with its connection $\nabla$ corresponds to $\mathcal{F}$. Hence we know that $M \otimes^\wedge_D \Omega_D^*$ computes the crystalline cohomology of $\mathcal{F}$, see Proposition \ref{proposition-compute-cohomology-crystal}. Hence it suffices to show that the base change maps (induced by $a$ and $b$) $$ M' \otimes^\wedge_{D'} \Omega^*_{D'} \longrightarrow M \otimes^\wedge_D \Omega^*_D \quad\text{and}\quad M \otimes^\wedge_D \Omega^*_D \longrightarrow M' \otimes^\wedge_{D'} \Omega^*_{D'} $$ are quasi-isomorphisms. Since $a \circ b = \text{id}_{D'}$ we see that the composition one way around is the identity on the complex $M' \otimes^\wedge_{D'} \Omega^*_{D'}$. Hence it suffices to show that the map $$ M \otimes^\wedge_D \Omega^*_D \longrightarrow M \otimes^\wedge_D \Omega^*_D $$ induced by $b \circ a : D \to D$ is a quasi-isomorphism. (Note that we have the same complex on both sides as $M = M' \otimes^\wedge_{D', b} D$, hence $M \otimes^\wedge_{D, b \circ a} D = M' \otimes^\wedge_{D', b \circ a \circ b} D = M' \otimes^\wedge_{D', b} D = M$.) In fact, we claim that for any divided power $A$-algebra homomorphism $\rho : D \to D$ compatible with the augmentation to $C$ the induced map $M \otimes^\wedge_D \Omega^*_D \to M \otimes^\wedge_{D, \rho} \Omega^*_D$ is a quasi-isomorphism. \medskip\noindent Write $\rho(x_i) = x_i + z_i$. The elements $z_i$ are in the divided power ideal of $D$ because $\rho$ is compatible with the augmentation to $C$. Hence we can factor the map $\rho$ as a composition $$ D \xrightarrow{\sigma} D\langle \xi_i \rangle^\wedge \xrightarrow{\tau} D $$ where the first map is given by $x_i \mapsto x_i + \xi_i$ and the second map is the divided power $D$-algebra map which maps $\xi_i$ to $z_i$. (This uses the universal properties of polynomial algebra, divided power polynomial algebras, divided power envelopes, and $p$-adic completion.) Note that there exists an {\it automorphism} $\alpha$ of $D\langle \xi_i \rangle^\wedge$ with $\alpha(x_i) = x_i - \xi_i$ and $\alpha(\xi_i) = \xi_i$. Applying Lemma \ref{lemma-relative-poincare} to $\alpha \circ \sigma$ (which maps $x_i$ to $x_i$) and using that $\alpha$ is an isomorphism we conclude that $\sigma$ induces a quasi-isomorphism of $M \otimes^\wedge_D \Omega^*_D$ with $M \otimes^\wedge_{D, \sigma} \Omega^*_{D\langle x_i \rangle^\wedge}$. On the other hand the map $\tau$ has as a left inverse the map $D \to D\langle x_i \rangle^\wedge$, $x_i \mapsto x_i$ and we conclude (using Lemma \ref{lemma-relative-poincare} once more) that $\tau$ induces a quasi-isomorphism of $M \otimes^\wedge_{D, \sigma} \Omega^*_{D\langle x_i \rangle^\wedge}$ with $M \otimes^\wedge_{D, \tau \circ \sigma} \Omega^*_D$. Composing these two quasi-isomorphisms we obtain that $\rho$ induces a quasi-isomorphism $M \otimes^\wedge_D \Omega^*_D \to M \otimes^\wedge_{D, \rho} \Omega^*_D$ as desired.
Note that $X' \to X$ is a homeomorphism hence we can identify the underlying topological spaces of $X$ and $X'$. The question is clearly local on $X$, hence we may assume $X$, $X'$, and $S$ affine and $X' \to X$ given as a composition $$ X' = X_n \to X_{n - 1} \to X_{n - 2} \to \ldots \to X_0 = X $$ where each morphism $X_{i + 1} \to X_i$ is an $\alpha_p$-cover. Denote $\mathcal{F}_i$ the pullback of $\mathcal{F}$ to $X_i$. It suffices to prove that each of the maps $$ R\Gamma(\text{Cris}(X_i/S), \mathcal{F}_i) \longrightarrow R\Gamma(\text{Cris}(X_{i + 1}/S), \mathcal{F}_{i + 1}) $$ fits into a triangle whose third member has cohomology groups annihilated by $p$. (This uses axiom TR4 for the triangulated category $D(X)$. Details omitted.) \medskip\noindent Hence we may assume that $S = \Spec(A)$, $X = \Spec(C)$, $X' = \Spec(C')$ and $C' = C[z]/(z^p - c)$ for some $c \in C$. Choose a polynomial algebra $P$ over $A$ and a surjection $P \to C$. Let $D$ be the $p$-adically completed divided power envelop of $\Ker(P \to C)$ in $P$ as in (\ref{equation-D}). Set $P' = P[z]$ with surjection $P' \to C'$ mapping $z$ to the class of $z$ in $C'$. Choose a lift $\lambda \in D$ of $c \in C$. Then we see that the $p$-adically completed divided power envelope $D'$ of $\Ker(P' \to C')$ in $P'$ is isomorphic to the $p$-adic completion of $D[z]\langle \xi \rangle/(\xi - (z^p - \lambda))$, see Lemma \ref{lemma-computation} and its proof. Thus we see that the result follows from this lemma by the computation of cohomology of crystals in quasi-coherent modules in Proposition \ref{proposition-compute-cohomology-crystal}.
It suffices to prove the first statement. To see this we may assume that $X$ is \'etale over $\mathbf{A}^d_{S_0}$, see Morphisms, Lemma \ref{morphisms-lemma-smooth-etale-over-affine-space}. Denote $\varphi : X \to \mathbf{A}^d_{S_0}$ this \'etale morphism. In this case the relative Frobenius of $X/S_0$ fits into a diagram $$ \xymatrix{ X \ar[d] \ar[r] & X^{(1)} \ar[d] \\ \mathbf{A}^d_{S_0} \ar[r] & \mathbf{A}^d_{S_0} } $$ where the lower horizontal arrow is the relative frobenius morphism of $\mathbf{A}^d_{S_0}$ over $S_0$. This is the morphism which raises all the coordinates to the $p$th power, hence it is an iterated $\alpha_p$-cover. The proof is finished by observing that the diagram is a fibre square, see \'Etale Morphisms, Lemma \ref{etale-lemma-relative-frobenius-etale}.
Let $(M, \nabla)$ be given. We are going to construct a crystal in quasi-coherent modules $\mathcal{F}$. Write $\nabla(m) = \sum \theta_i(m)\text{d}x_i$. Then $\theta_i \circ \theta_j = \theta_j \circ \theta_i$ and we can set $\theta_K(m) = (\prod \theta_i^{k_i})(m)$ for any multi-index $K = (k_i)$ with $k_i \geq 0$ and $\sum k_i < \infty$. \medskip\noindent Let $(U, T, \delta)$ be any object of $\text{Cris}(X/S)$ with $T$ affine. Say $T = \Spec(B)$ and the ideal of $U \to T$ is $J_B \subset B$. By Lemma \ref{lemma-set-generators} there exists an integer $e$ and a morphism $$ f : (U, T, \delta) \longrightarrow (X, T_e, \bar\gamma) $$ where $T_e = \Spec(D_e)$ as in the proof of Lemma \ref{lemma-crystals-on-affine}. Choose such an $e$ and $f$; denote $f : D \to B$ also the corresponding divided power $A$-algebra map. We will set $\mathcal{F}_T$ equal to the quasi-coherent sheaf of $\mathcal{O}_T$-modules associated to the $B$-module $$ M \otimes_{D, f} B. $$ However, we have to show that this is independent of the choice of $f$. Suppose that $g : D \to B$ is a second such morphism. Since $f$ and $g$ are morphisms in $\text{Cris}(X/S)$ we see that the image of $f - g : D \to B$ is contained in the divided power ideal $J_B$. Write $\xi_i = f(x_i) - g(x_i) \in J_B$. By analogy with the proof of Lemma \ref{lemma-crystals-on-affine} we define an isomorphism $$ c_{f, g} : M \otimes_{D, f} B \longrightarrow M \otimes_{D, g} B $$ by the formula $$ m \otimes 1 \longmapsto  \sum\nolimits_K \theta_K(m) \otimes \prod \xi_i^{[k_i]} $$ which makes sense by our remarks above and the fact that $\nabla$ is topologically quasi-nilpotent (so the sum is finite!). A computation shows that $$ c_{g, h} \circ c_{f, g} = c_{f, h} $$ if given a third morphism $h : (U, T, \delta) \longrightarrow (X, T_e, \bar\gamma)$. It is also true that $c_{f, f} = 1$. Hence these maps are all isomorphisms and we see that the module $\mathcal{F}_T$ is independent of the choice of $f$. \medskip\noindent If $a : (U', T', \delta') \to (U, T, \delta)$ is a morphism of affine objects of $\text{Cris}(X/S)$, then choosing $f' = f \circ a$ it is clear that there exists a canonical isomorphism $a^*\mathcal{F}_T \to \mathcal{F}_{T'}$. We omit the verification that this map is independent of the choice of $f$. Using these maps as the restriction maps it is clear that we obtain a crystal in quasi-coherent modules on the full subcategory of $\text{Cris}(X/S)$ consisting of affine objects. We omit the proof that this extends to a crystal on all of $\text{Cris}(X/S)$. We also omit the proof that this procedure is a functor and that it is quasi-inverse to the functor constructed in Lemma \ref{lemma-crystals-on-affine}.
Using assumption (1) and Lemma \ref{lemma-compare} we see that $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is isomorphic to $R\Gamma(\mathcal{C}, \mathcal{F})$. Note that the categories $\mathcal{C}$ used in Lemmas \ref{lemma-compare} and \ref{lemma-complete} agree. Let $f : T \to T'$ be a closed immersion as in (2). Surjectivity of $c_f : f^*\mathcal{F}_{T'} \to \mathcal{F}_T$ is equivalent to surjectivity of $\mathcal{F}_{T'} \to f_*\mathcal{F}_T$. Hence, if $\mathcal{F}$ satisfies (1) and (2), then we obtain a short exact sequence $$ 0 \to \mathcal{K} \to \mathcal{F}_{T'} \to f_*\mathcal{F}_T \to 0 $$ of quasi-coherent $\mathcal{O}_{T'}$-modules on $T'$, see Schemes, Section \ref{schemes-section-quasi-coherent} and in particular Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. Thus, if $T'$ is affine, then we conclude that the restriction map $\mathcal{F}(U', T', \delta') \to \mathcal{F}(U, T, \delta)$ is surjective by the vanishing of $H^1(T', \mathcal{K})$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. Hence the transition maps of the inverse systems in Lemma \ref{lemma-complete} are surjective. We conclude that $R^pg_*(\mathcal{F}|_\mathcal{C}) = 0$ for all $p \geq 1$ where $g$ is as in Lemma \ref{lemma-complete}. The object $D$ of the category $\mathcal{C}^\wedge$ satisfies the assumption of Lemma \ref{lemma-category-with-covering} by Lemma \ref{lemma-generator-completion} with $$ D \times \ldots \times D = D(n) $$ in $\mathcal{C}$ because $D(n)$ is the $n + 1$-fold coproduct of $D$ in $\text{Cris}^\wedge(C/A)$, see Lemma \ref{lemma-property-Dn}. Thus we win.
We will prove this using the two spectral sequences associated to the double complex $K^{*, *}$ with terms $$ K^{a, b} = M \otimes_D^\wedge \Omega^a_{D(b)} $$ What do we know so far? Well, Lemma \ref{lemma-vanishing} tells us that each column $K^{a, *}$, $a > 0$ is acyclic. Proposition \ref{proposition-compute-cohomology} tells us that the first column $K^{0, *}$ is quasi-isomorphic to $R\Gamma(\text{Cris}(X/S), \mathcal{F})$. Hence the first spectral sequence associated to the double complex shows that there is a canonical quasi-isomorphism of $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ with $\text{Tot}(K^{*, *})$. \medskip\noindent Next, let's consider the rows $K^{*, b}$. By Lemma \ref{lemma-structure-Dn} each of the $b + 1$ maps $D \to D(b)$ presents $D(b)$ as the $p$-adic completion of a divided power polynomial algebra over $D$. Hence Lemma \ref{lemma-relative-poincare} shows that the map $$ M \otimes^\wedge_D\Omega^*_D \longrightarrow M \otimes^\wedge_{D(b)} \Omega^*_{D(b)} = K^{*, b} $$ is a quasi-isomorphism. Note that each of these maps defines the {\it same} map on cohomology (and even the same map in the derived category) as the inverse is given by the co-diagonal map $D(b) \to D$ (corresponding to the multiplication map $P \otimes_A \ldots \otimes_A P \to P$). Hence if we look at the $E_1$ page of the second spectral sequence we obtain $$ E_1^{a, b} = H^a(M \otimes^\wedge_D\Omega^*_D) $$ with differentials $$ E_1^{a, 0} \xrightarrow{0} E_1^{a, 1} \xrightarrow{1} E_1^{a, 2} \xrightarrow{0} E_1^{a, 3} \xrightarrow{1} \ldots $$ as each of these is the alternation sum of the given identifications $H^a(M \otimes^\wedge_D\Omega^*_D) = E_1^{a, 0} = E_1^{a, 1} = \ldots$. Thus we see that the $E_2$ page is equal $H^a(M \otimes^\wedge_D\Omega^*_D)$ on the first row and zero elsewhere. It follows that the identification of $M \otimes^\wedge_D\Omega^*_D$ with the first row induces a quasi-isomorphism of $M \otimes^\wedge_D\Omega^*_D$ with $\text{Tot}(K^{*, *})$.
By Lemma \ref{lemma-automatic-connection} we get a de Rham complex as indicated in the lemma. We abbreviate $\mathcal{H} = \mathcal{F} \otimes \Omega^i_{X/S}$. Let $X' \subset X$ be an affine open subscheme which maps into an affine open subscheme $S' \subset S$. Then $$ (Ru_{X/S, *}\mathcal{H})|_{X'_{Zar}} = Ru_{X'/S', *}(\mathcal{H}|_{\text{Cris}(X'/S')}), $$ see Lemma \ref{lemma-localize}. Thus Lemma \ref{lemma-cohomology-is-zero} shows that $Ru_{X/S, *}\mathcal{H}$ is a complex of sheaves on $X_{Zar}$ whose cohomology on any affine open is trivial. As $X$ has a basis for its topology consisting of affine opens this implies that $Ru_{X/S, *}\mathcal{H}$ is quasi-isomorphic to zero.
Namely, for an algebraic space $Y$ over $B$ a morphism $X \to Y$ over $B$ is the same thing as a collection of morphism $U_a \to Y$ which agree on the overlaps $U_a \times_X U_b$ for all $a, b \in A$, see Descent on Spaces, Lemma \ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}.
We encourage the reader to look instead at Lemma \ref{lemma-colimit-check-etale-locally} and its proof. \medskip\noindent Let $Z$ be an algebraic space over $B$. Suppose that $f_i : X_i \to Z$ is a family of morphisms such that for each $i \to j$ the composition $X_i \to X_j \to Z$ is equal to $f_i$. We have to construct a morphism of algebraic spaces $f : X \to Z$ over $B$ such that we can recover $f_i$ as the composition $X_i \to X \to Z$. Let $W \to Z$ be a surjective \'etale morphism of a scheme to $Z$. We may assume that $W$ is a disjoint union of affines and in particular we may assume that $W \to Z$ is separated and that $W$ is separated over $B$. For each $i$ set $U_i = W \times_{Z, f_i} X_i$ and denote $h_i : U_i \to W$ the projection. Then $U_i \to X_i$ forms an object of $\lim X_{i, spaces, \etale}$ with $U_i \to X_i$ separated. By assumption (5) we can find a separated \'etale morphism $U \to X$ of algebraic spaces and (functorial) isomorphisms $U_i = X_i \times_X U$. By assumption (4) there exists a morphism $h : U \to W$ over $B$ such that the compositions $U_i \to U \to W$ are $h_i$. Let $g : U \to Z$ be the composition of $h$ with the map $W \to Z$. To finish the proof we have to show that $g : U \to Z$ descends to a morphism $X \to Z$. To do this, consider the morphism $(h, h) : U \times_X U \to W \times_S W$. Composing with $U_i \times_{X_i} U_i \to U \times_X U$ we obtain $(h_i, h_i)$ which factors through $W \times_Z W$. Since $U \times_X U$ is the colimit of the algebraic spaces $U_i \times_{X_i} U_i$ in the category of algebraic spaces separated over $B$ by (4) we see that $(h, h)$ factors through $W \times_Z W$. Hence the two compositions $U \times_X U \to U \to W \to Z$ are equal. Because each $U_i \to X_i$ is surjective and assumption (2) we see that $U \to X$ is surjective. As $Z$ is a sheaf for the \'etale topology, we conclude that $g : U \to Z$ descends to $f : X \to Z$ as desired.
Each of the properties of morphisms of algebraic spaces mentioned in the statement of the lemma is preserved by arbitrary base change, see the lists in Spaces, Section \ref{spaces-section-lists}. Thus we can apply Lemma \ref{lemma-reduce-to-scheme-base} to see that we can work \'etale locally on $Y$. In this way we reduce to the case where $Y$ is a scheme; some details omitted. In this case $X$ is also a scheme and the result follows from \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-glue-etale-sheaf-integral-surjective}, \ref{etale-cohomology-lemma-glue-etale-sheaf-proper-surjective}, or \ref{etale-cohomology-lemma-glue-etale-sheaf-fppf-cover}.
Observe that the base change of a proper surjective morphism is proper and surjective, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-proper} and \ref{spaces-morphisms-lemma-base-change-surjective}. Hence by Lemma \ref{lemma-reduce-to-scheme-base} we may work \'etale locally on $Y$. Hence we reduce to $Y$ being an affine scheme; some details omitted. \medskip\noindent Assume $Y$ is affine. By Lemma \ref{lemma-reduce-to-scheme-source} it suffices to find a morphism $X' \to X$ where $X'$ is a scheme such that $X' \to Y$ is surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. \medskip\noindent In case $X \to Y$ is integral and surjective, we can take $X = X'$ as an integral morphism is representable. \medskip\noindent If $f$ is proper and surjective, then the algebraic space $X$ is quasi-compact and separated, see Morphisms of Spaces, Section \ref{spaces-morphisms-section-quasi-compact} and Lemma \ref{spaces-morphisms-lemma-separated-over-separated}. Choose a scheme $X'$ and a surjective finite morphism $X' \to X$, see Limits of Spaces, Proposition \ref{spaces-limits-proposition-there-is-a-scheme-finite-over}. Then $X' \to Y$ is surjective and proper. \medskip\noindent Finally, if $X \to Y$ is surjective and flat and locally of finite presentation then we can take an affine \'etale covering $\{U_i \to X\}$ and set $X'$ equal to the disjoint $\coprod U_i$.
Observe that $X = Y' \coprod Z \to Y$ is a proper surjective morphism. Thus it suffice to construct an equivalence of categories $$ \Sh(Y'_\etale) \times_{\Sh(E_\etale)} \Sh(Z_\etale) \longrightarrow \text{descent data for \'etale sheaves wrt }\{X \to Y\} $$ compatible with pullback functors from $Y$ because then we can use Lemma \ref{lemma-glue-etale-sheaf-proper-surjective} to conclude. Thus let $(\mathcal{G}', \mathcal{G}, \alpha)$ be an object of $\Sh(Y'_\etale) \times_{\Sh(E_\etale)} \Sh(Z_\etale)$ with notation as in Categories, Example \ref{categories-example-2-fibre-product-categories}. Then we can consider the sheaf $\mathcal{F}$ on $X$ defined by taking $\mathcal{G}'$ on the summand $Y'$ and $\mathcal{G}$ on the summand $Z$. We have $$ X \times_Y X = Y' \times_Y Y' \amalg Y' \times_Y Z \amalg Z \times_Y Y' \amalg Z \times_Y Z = Y' \times_Y Y' \amalg E \amalg E \amalg Z $$ The isomorphisms of the two pullbacks of $\mathcal{F}$ to this algebraic space are obvious over the summands $E$, $E$, $Z$. The interesting part of the proof is to find an isomorphism $\text{pr}_{0, small}^{-1}\mathcal{G}' \to \text{pr}_{1, small}^{-1}\mathcal{G}'$ over $Y' \times_Y Y'$ satisfying the cocycle condition. However, our assumption that $Y' \to Y$ is an isomorphism over $Y \setminus Z$ implies that $$ h : Y \coprod E \times_Z E \longrightarrow Y' \times_Y Y' $$ is a surjective proper morphism. (It is in fact a finite morphism as it is the disjoint union of two closed immersions.) Hence it suffices to construct an isomorphism of the pullbacks of $\text{pr}_{0, small}^{-1}\mathcal{G}'$and $\text{pr}_{1, small}^{-1}\mathcal{G}'$ by $h_{small}$ satisfying a certain cocycle condition. For the diagonal, it is clear how to do this. And for the pullback to $E \times_Z E$ we use that both sheaves pull back to the pullback of $\mathcal{G}$ by the morphism $E \times_Z E \to Z$. We omit the details.
Recall that $U$ gives rise to a representable sheaf $\mathcal{F} = h_U$ in $\Sh(X_{spaces, \etale}) = \Sh(X_\etale)$, see Properties of Spaces, Section \ref{spaces-properties-section-etale-site}. The descent datum on $U$ relative to $f$ exactly gives a descent datum $(\mathcal{F}, \varphi)$ for \'etale sheaves with respect to $\{X \to Y\}$. By Lemma \ref{lemma-glue-etale-sheaf-proper-surjective} this descent datum is effective. Let $\mathcal{G}$ be the corresponding sheaf on $Y_\etale$. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-sheaf-gives-space} we obtain an \'etale morphism $V \to Y$ of algebraic spaces corresponding to $\mathcal{G}$; we omit the verification of the set theoretic condition\footnote{It follows from the fact that $\mathcal{F}$ satisfies the corresponding condition.}. The given isomorphism $\mathcal{F} \to f_{small}^{-1}\mathcal{G}$ corresponds to an isomorphism $U \to V \times_Y X$ compatible with the descent datum.
Let $(V' \to Y', W \to Z, \alpha)$ be an object of the right hand side. Recall that $V'$, resp.\ $W$ gives rise to a representable sheaf $\mathcal{G}' = h_{V'}$ in $\Sh(Y'_{spaces, \etale}) = \Sh(Y'_\etale)$, resp.\ $\mathcal{G} = h_W$  in $\Sh(Z_{spaces, \etale}) = \Sh(Z_\etale)$, see Properties of Spaces, Section \ref{spaces-properties-section-etale-site}. The isomorphism $\alpha : V' \times_{Y'} E \to W \times_Z E$ determines an isomorphism $j_{small}^{-1}\mathcal{G}' \to g_{small}^{-1}\mathcal{G}$ of sheaves on $E$. By Lemma \ref{lemma-glue-etale-sheaf-modification} we obtain a unique sheaf $\mathcal{F}$ on $Y$ pulling pack to $\mathcal{G}'$ and $\mathcal{G}$ compatibly with the isomorphism. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-sheaf-gives-space} we obtain an \'etale morphism $V \to Y$ of algebraic spaces corresponding to $\mathcal{F}$; we omit the verification of the set theoretic condition\footnote{It follows from the fact that $\mathcal{G}$ and $\mathcal{G}'$ satisfies the corresponding condition.}. The given isomorphism $\mathcal{G}' \to f_{small}^{-1}\mathcal{F}$ and $\mathcal{G} \to i_{small}^{-1}\mathcal{F}$ corresponds to isomorphisms $V' \to V \times_Y Y'$ and $W \to V \times_Y Z$ compatible with $\alpha$ as desired.