hcju/stacksqa huggingface dataset:
Since flatness is defined in terms of the maps of local rings (Morphisms, Definition \ref{morphisms-definition-flat}) what has to be shown is the following algebraic fact: Suppose $A \to B \to C$ are local homomorphisms of local rings, and assume $B \to C$ is flat. Then $A \to B$ is flat if and only if $A \to C$ is flat. If $A \to B$ is flat, then $A \to C$ is flat by Algebra, Lemma \ref{algebra-lemma-composition-flat}. Conversely, assume $A \to C$ is flat. Note that $B \to C$ is faithfully flat, see Algebra, Lemma \ref{algebra-lemma-local-flat-ff}. Hence $A \to B$ is flat by Algebra, Lemma \ref{algebra-lemma-flat-permanence}. (Also see Morphisms, Lemma \ref{morphisms-lemma-flat-permanence} for a direct proof.)
Being locally of finite presentation is Zariski local on the source and the target, see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-presentation-characterize}. It is a property which is preserved under composition, see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-presentation}. This proves (1), (2) and (3) of Lemma \ref{lemma-properties-morphisms-local-source}. The final condition (4) is Lemma \ref{lemma-flat-finitely-presented-permanence-algebra}. Hence we win.
Being locally of finite type is Zariski local on the source and the target, see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-type-characterize}. It is a property which is preserved under composition, see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-type}, and a flat morphism locally of finite presentation is locally of finite type, see Morphisms, Lemma \ref{morphisms-lemma-finite-presentation-finite-type}. This proves (1), (2) and (3) of Lemma \ref{lemma-properties-morphisms-local-source}. The final condition (4) is Lemma \ref{lemma-finite-type-local-source-fppf-algebra}. Hence we win.
Being an open morphism is clearly Zariski local on the source and the target. It is a property which is preserved under composition, see Morphisms, Lemma \ref{morphisms-lemma-composition-open}, and a flat morphism of finite presentation is open, see Morphisms, Lemma \ref{morphisms-lemma-fppf-open} This proves (1), (2) and (3) of Lemma \ref{lemma-properties-morphisms-local-source}. The final condition (4) follows from Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}. Hence we win.
Let $f : X \to Y$ be a morphism of schemes. Let $\{X_i \to X\}_{i \in I}$ be an fppf covering. Denote $f_i : X_i \to X$ the compositions. We have to show that $f$ is universally open if and only if each $f_i$ is universally open. If $f$ is universally open, then also each $f_i$ is universally open since the maps $X_i \to X$ are universally open and compositions of universally open morphisms are universally open (Morphisms, Lemmas \ref{morphisms-lemma-fppf-open} and \ref{morphisms-lemma-composition-open}). Conversely, assume each $f_i$ is universally open. Let $Y' \to Y$ be a morphism of schemes. Denote $X' = Y' \times_Y X$ and $X'_i = Y' \times_Y X_i$. Note that $\{X_i' \to X'\}_{i \in I}$ is an fppf covering also. The morphisms $f'_i : X_i' \to Y'$ are open by assumption. Hence by the Lemma \ref{lemma-open-fppf-local-source} above we conclude that $f' : X' \to Y'$ is open as desired.
Combine Lemma \ref{lemma-properties-morphisms-local-source} with Morphisms, Lemma \ref{morphisms-lemma-syntomic-characterize} (local for Zariski on source and target), Morphisms, Lemma \ref{morphisms-lemma-composition-syntomic} (pre-composing), and Lemma \ref{lemma-syntomic-smooth-etale-permanence} (part (4)).
Combine Lemma \ref{lemma-properties-morphisms-local-source} with Morphisms, Lemma \ref{morphisms-lemma-smooth-characterize} (local for Zariski on source and target), Morphisms, Lemma \ref{morphisms-lemma-composition-smooth} (pre-composing), and Lemma \ref{lemma-syntomic-smooth-etale-permanence} (part (4)).
Combine Lemma \ref{lemma-properties-morphisms-local-source} with Morphisms, Lemma \ref{morphisms-lemma-etale-characterize} (local for Zariski on source and target), Morphisms, Lemma \ref{morphisms-lemma-composition-etale} (pre-composing), and Lemma \ref{lemma-syntomic-smooth-etale-permanence} (part (4)).
We are going to use Lemma \ref{lemma-properties-morphisms-local-source}. By Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-characterize} the property of being locally quasi-finite is local for Zariski on source and target. By Morphisms, Lemmas \ref{morphisms-lemma-composition-quasi-finite} and \ref{morphisms-lemma-etale-locally-quasi-finite} we see the precomposition of a locally quasi-finite morphism by an \'etale morphism is locally quasi-finite. Finally, suppose that $X \to Y$ is a morphism of affine schemes and that $X' \to X$ is a surjective \'etale morphism of affine schemes such that $X' \to Y$ is locally quasi-finite. Then $X' \to Y$ is of finite type, and by Lemma \ref{lemma-finite-type-local-source-fppf-algebra} we see that $X \to Y$ is of finite type also. Moreover, by assumption $X' \to Y$ has finite fibres, and hence $X \to Y$ has finite fibres also. We conclude that $X \to Y$ is quasi-finite by Morphisms, Lemma \ref{morphisms-lemma-quasi-finite}. This proves the last assumption of Lemma \ref{lemma-properties-morphisms-local-source} and finishes the proof.
We are going to use Lemma \ref{lemma-properties-morphisms-local-source}. By Morphisms, Lemma \ref{morphisms-lemma-unramified-characterize} the property of being unramified (resp.\ G-unramified) is local for Zariski on source and target. By Morphisms, Lemmas \ref{morphisms-lemma-composition-unramified} and \ref{morphisms-lemma-etale-smooth-unramified} we see the precomposition of an unramified (resp.\ G-unramified) morphism by an \'etale morphism is unramified (resp.\ G-unramified). Finally, suppose that $X \to Y$ is a morphism of affine schemes and that $f : X' \to X$ is a surjective \'etale morphism of affine schemes such that $X' \to Y$ is unramified (resp.\ G-unramified). Then $X' \to Y$ is of finite type (resp.\ finite presentation), and by Lemma \ref{lemma-finite-type-local-source-fppf-algebra} (resp.\ Lemma \ref{lemma-flat-finitely-presented-permanence-algebra}) we see that $X \to Y$ is of finite type (resp.\ finite presentation) also. By Morphisms, Lemma \ref{morphisms-lemma-triangle-differentials-smooth} we have a short exact sequence $$ 0 \to f^*\Omega_{X/Y} \to \Omega_{X'/Y} \to \Omega_{X'/X} \to 0. $$ As $X' \to Y$ is unramified we see that the middle term is zero. Hence, as $f$ is faithfully flat we see that $\Omega_{X/Y} = 0$. Hence $X \to Y$ is unramified (resp.\ G-unramified), see Morphisms, Lemma \ref{morphisms-lemma-unramified-omega-zero}. This proves the last assumption of Lemma \ref{lemma-properties-morphisms-local-source} and finishes the proof.
We write everything out completely. \medskip\noindent Proof of (1). Let $f : X \to Y$ be a morphism of schemes. Let $\{X_i \to X\}_{i \in I}$ be an \'etale covering of $X$. If each composition $h_i : X_i \to Y$ has $\mathcal{P}$, then for each $x \in X$ we can find an $i \in I$ and a point $x_i \in X_i$ mapping to $x$. Then $(X_i, x_i) \to (X, x)$ is an \'etale morphism of germs, and $\text{id}_Y : Y \to Y$ is an \'etale morphism, and $h_i$ is as in part (3) of Definition \ref{definition-local-source-target}. Thus we see that $f$ has $\mathcal{P}$. Conversely, if $f$ has $\mathcal{P}$ then each $X_i \to Y$ has $\mathcal{P}$ by Definition \ref{definition-local-source-target} part (1). \medskip\noindent Proof of (2). Let $f : X \to Y$ be a morphism of schemes. Let $\{Y_i \to Y\}_{i \in I}$ be an \'etale covering of $Y$. Write $X_i = Y_i \times_Y X$ and $h_i : X_i \to Y_i$ for the base change of $f$.  If each  $h_i : X_i \to Y_i$ has $\mathcal{P}$, then for each $x \in X$ we pick an $i \in I$ and a point $x_i \in X_i$ mapping to $x$. Then $(X_i, x_i) \to (X, x)$ is an \'etale morphism of germs, $Y_i \to Y$ is \'etale, and $h_i$ is as in part (3) of Definition \ref{definition-local-source-target}. Thus we see that $f$ has $\mathcal{P}$. Conversely, if $f$ has $\mathcal{P}$, then each $X_i \to Y_i$ has $\mathcal{P}$ by Definition \ref{definition-local-source-target} part (2). \medskip\noindent Proof of (3). Assume $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is \'etale. For every $x \in X$ we can think of $(X, x) \to (X, x)$ as an \'etale morphism of germs, $Y \to Z$ is an \'etale morphism, and $h = f$ is as in part (3) of Definition \ref{definition-local-source-target}. Thus we see that $g \circ f$ has $\mathcal{P}$. \medskip\noindent Proof of (4). Let $f : X \to Y$ be a morphism and $g : Y \to Z$ \'etale such that $g \circ f$ has $\mathcal{P}$. Then by Definition \ref{definition-local-source-target} part (2) we see that $\text{pr}_Y : Y \times_Z X \to Y$ has $\mathcal{P}$. But the morphism $(f, 1) : X \to Y \times_Z X$ is \'etale as a section to the \'etale projection $\text{pr}_X : Y \times_Z X \to X$, see Morphisms, Lemma \ref{morphisms-lemma-etale-permanence}. Hence $f = \text{pr}_Y \circ (f, 1)$ has $\mathcal{P}$ by Definition \ref{definition-local-source-target} part (1).
The equivalence of (a) and (b) is part of Definition \ref{definition-local-source-target}. The equivalence of (a) and (e) is Lemma \ref{lemma-local-source-target-implies} part (2). The equivalence of (a) and (f) is Lemma \ref{lemma-local-source-target-implies} part (1). As (a) is now equivalent to (e) and (f) it follows that (a) equivalent to (g). \medskip\noindent It is clear that (c) implies (a). If (a) holds, then for any diagram as in (c) the morphism $f \circ a$ has $\mathcal{P}$ by Definition \ref{definition-local-source-target} part (1), whereupon $h$ has $\mathcal{P}$ by Lemma \ref{lemma-local-source-target-implies} part (4). Thus (a) and (c) are equivalent. It is clear that (c) implies (d). To see that (d) implies (a) assume we have a diagram as in (c) with $a : U \to X$ surjective and $h$ having $\mathcal{P}$. Then $b \circ h$ has $\mathcal{P}$ by Lemma \ref{lemma-local-source-target-implies} part (3). Since $\{a : U \to X\}$ is an \'etale covering we conclude that $f$ has $\mathcal{P}$ by Lemma \ref{lemma-local-source-target-implies} part (1).
Lemma \ref{lemma-largest-open-of-the-source} applies since $\mathcal{P}$ is \'etale local on the source by Lemma \ref{lemma-local-source-target-implies}. \medskip\noindent Assume $x \in W(f)$. Let $U' \subset X'$ and $V' \subset Y'$ be open neighbourhoods of $x'$ and $y'$ such that $f'(U') \subset V'$, $g'(U') \subset W(f)$ and $g'|_{U'}$ and $g|_{V'}$ are \'etale. Then $f \circ g'|_{U'} = g \circ f'|_{U'}$ has $\mathcal{P}$ by property (1) of Definition \ref{definition-local-source-target}. Then $f'|_{U'} : U' \to V'$ has property $\mathcal{P}$ by (4) of Lemma \ref{lemma-local-source-target-implies}. Then by (3) of Lemma \ref{lemma-local-source-target-implies} we conclude that $f'_{U'} : U' \to Y'$ has $\mathcal{P}$. Hence $U' \subset W(f')$ by definition. Hence $x' \in W(f')$. \medskip\noindent Assume $x' \in W(f')$. Let $U' \subset X'$ and $V' \subset Y'$ be open neighbourhoods of $x'$ and $y'$ such that $f'(U') \subset V'$, $U' \subset W(f')$ and $g'|_{U'}$ and $g|_{V'}$ are \'etale. Then $U' \to Y'$ has $\mathcal{P}$ by definition of $W(f')$. Then $U' \to V'$ has $\mathcal{P}$ by (4) of Lemma \ref{lemma-local-source-target-implies}. Then $U' \to Y$ has $\mathcal{P}$ by (3) of Lemma \ref{lemma-local-source-target-implies}. Let $U \subset X$ be the image of the \'etale (hence open) morphism $g'|_U' : U' \to X$. Then $\{U' \to U\}$ is an \'etale covering and we conclude that $U \to Y$ has $\mathcal{P}$ by (1) of Lemma \ref{lemma-local-source-target-implies}. Thus $U \subset W(f)$ by definition. Hence $x \in W(f)$.
If a morphism of germs $(X, x) \to (S, s)$ has $\mathcal{Q}$, then there are arbitrarily small neighbourhoods $U \subset X$ of $x$ and $V \subset S$ of $s$ such that a representative $U \to V$ of $(X, x) \to (S, s)$ has $\mathcal{P}$. This follows from Lemma \ref{lemma-local-source-target-implies}. Let $$ \xymatrix{ (U', u') \ar[r]_{h'} \ar[d]_a & (V', v') \ar[d]^b \\ (U, u) \ar[r]^h & (V, v) } $$ be as in Definition \ref{definition-local-source-target-at-point}. Choose $U_1 \subset U$ and a representative $h_1 : U_1 \to V$ of $h$. Choose $V'_1 \subset V'$ and an \'etale representative $b_1 : V'_1 \to V$ of $b$ (Definition \ref{definition-etale-morphism-germs}). Choose $U'_1 \subset U'$ and representatives $a_1 : U'_1 \to U_1$ and $h'_1 : U'_1 \to V'_1$ of $a$ and $h'$ with $a_1$ \'etale. After shrinking $U'_1$ we may assume $h_1 \circ a_1 = b_1 \circ h'_1$. By the initial remark of the proof, we are trying to show $u' \in W(h'_1) \Leftrightarrow u \in W(h_1)$ where $W(-)$ is as in Lemma \ref{lemma-largest-open-of-the-source}. Thus the lemma follows from Lemma \ref{lemma-etale-etale-local-source-target}.
Given a diagram as in Definition \ref{definition-local-source-target-at-point} we obtain the following diagram of local homomorphisms of local rings $$ \xymatrix{ \mathcal{O}_{U', u'} & \mathcal{O}_{V', v'} \ar[l] \\ \mathcal{O}_{U, u} \ar[u] & \mathcal{O}_{V, v} \ar[l] \ar[u] } $$ Note that the vertical arrows are localizations of \'etale ring maps, in particular they are essentially of finite presentation, flat, and unramified (see Algebra, Section \ref{algebra-section-etale}). In particular the vertical maps are faithfully flat, see Algebra, Lemma \ref{algebra-lemma-local-flat-ff}. Now, if the upper horizontal arrow is flat, then the lower horizontal arrow is flat by an application of Algebra, Lemma \ref{algebra-lemma-flat-permanence} with $R = \mathcal{O}_{V, v}$, $S = \mathcal{O}_{U, u}$ and $M = \mathcal{O}_{U', u'}$. If the lower horizontal arrow is flat, then the ring map $$ \mathcal{O}_{V', v'} \otimes_{\mathcal{O}_{V, v}} \mathcal{O}_{U, u} \longleftarrow \mathcal{O}_{V', v'} $$ is flat by Algebra, Lemma \ref{algebra-lemma-flat-base-change}. And the ring map $$ \mathcal{O}_{U', u'} \longleftarrow \mathcal{O}_{V', v'} \otimes_{\mathcal{O}_{V, v}} \mathcal{O}_{U, u} $$ is a localization of a map between \'etale ring extensions of $\mathcal{O}_{U, u}$, hence flat by Algebra, Lemma \ref{algebra-lemma-map-between-etale}.
Note that $U'_v \to U_v$ is \'etale as a base change of the \'etale morphism $U' \to U$. The scheme $U'_v$ is a scheme over $V'_v$. By Morphisms, Lemma \ref{morphisms-lemma-etale-over-field} the scheme $V'_v$ is a disjoint union of spectra of finite separable field extensions of $\kappa(v)$. One of these is $v' = \Spec(\kappa(v'))$. Hence $U'_{v'}$ is an open and closed subscheme of $U'_v$ and it follows that $U'_{v'} \to U'_v \to U_v$ is \'etale (as a composition of an open immersion and an \'etale morphism, see Morphisms, Section \ref{morphisms-section-etale}).
Given a diagram as in Definition \ref{definition-local-source-target-at-point} we obtain the following diagram of local homomorphisms of local rings $$ \xymatrix{ \mathcal{O}_{U', u'} & \mathcal{O}_{V', v'} \ar[l] \\ \mathcal{O}_{U, u} \ar[u] & \mathcal{O}_{V, v} \ar[l] \ar[u] } $$ Note that the vertical arrows are localizations of \'etale ring maps, in particular they are unramified (see Algebra, Section \ref{algebra-section-etale}). Hence $\kappa(u) \subset \kappa(u')$ and $\kappa(v) \subset \kappa(v')$ are finite separable field extensions. Thus we have $\text{trdeg}_{\kappa(v)} \kappa(u) = \text{trdeg}_{\kappa(v')} \kappa(u)$ which proves the lemma.
Given a diagram as in Definition \ref{definition-local-source-target-at-point} we obtain an \'etale morphism of fibres $U'_{v'} \to U_v$ mapping $u'$ to $u$, see Lemma \ref{lemma-etale-on-fiber}. Hence now the equality $\dim_u(U_v) = \dim_{u'}(U'_{v'})$ follows from Lemma \ref{lemma-dimension-at-point-local}.
We omit the proof of (1), but we remark that the morphism $\varphi'$ is the morphism $(f \times f)^*\varphi$ in the notation introduced in Remark \ref{remark-easier}. For (2) we indicate which morphism $f_0^*V \to f_1^*V$ gives the functorial isomorphism. Namely, since $f_0$ and $f_1$ both fit into the commutative diagram we see there is a unique morphism $r : X' \to X \times_S X$ with $f_i = \text{pr}_i \circ r$. Then we take \begin{eqnarray*} f_0^*V & = & X' \times_{f_0, X} V \\ & = & X' \times_{\text{pr}_0 \circ r, X} V \\ & = & X' \times_{r, X \times_S X} (X \times_S X) \times_{\text{pr}_0, X} V \\ & \xrightarrow{\varphi} & X' \times_{r, X \times_S X} (X \times_S X) \times_{\text{pr}_1, X} V \\ & = & X' \times_{\text{pr}_1 \circ r, X} V \\ & = & X' \times_{f_1, X} V \\ & = & f_1^*V \end{eqnarray*} We omit the verification that this works.
Suppose we have $h : X' \to X$ surjective and flat and $a, b : X \to Y$ morphisms such that $a \circ h = b \circ h$. As $h$ is surjective we see that $a$ and $b$ agree on underlying topological spaces. Pick $x' \in X'$ and set $x = h(x')$ and $y = a(x) = b(x)$. Consider the local ring maps $$ a^\sharp_x, b^\sharp_x : \mathcal{O}_{Y, y} \to \mathcal{O}_{X, x} $$ These become equal when composed with the flat local homomorphism $h^\sharp_{x'} : \mathcal{O}_{X, x} \to \mathcal{O}_{X', x'}$. Since a flat local homomorphism is faithfully flat (Algebra, Lemma \ref{algebra-lemma-local-flat-ff}) we conclude that $h^\sharp_{x'}$ is injective. Hence $a^\sharp_x = b^\sharp_x$ which implies $a = b$ as desired.
Let $X_1$, $X_2$ be schemes over $S$. Let $\alpha, \beta : X_2 \to X_1$ be morphisms over $S$. If $\alpha$, $\beta$ base change to the same morphism then we get a commutative diagram as follows $$ \xymatrix{ X_2 \ar[d]^\alpha & S' \times_S X_2 \ar[l] \ar[d] \ar[r] & X_2 \ar[d]^\beta \\ X_1 & S' \times_S X_1 \ar[l] \ar[r] & X_1 } $$ Hence it suffices to show that $S' \times_S X_2 \to X_2$ is an epimorphism. As the base change of a surjective and flat morphism it is surjective and flat (see Morphisms, Lemmas \ref{morphisms-lemma-base-change-surjective} and \ref{morphisms-lemma-base-change-flat}). Hence the lemma follows from Lemma \ref{lemma-surjective-flat-epi}.
Let $(V_i, \varphi_i)$, $i = 1, 2$ be descent data for $X \to S$. Let $\alpha, \beta : V_1 \to V_2$ be morphisms of descent data. Suppose that $f^*\alpha = f^*\beta$. Our task is to show that $\alpha = \beta$. Note that $\alpha$, $\beta$ are morphisms of schemes over $X$, and that $f^*\alpha$, $f^*\beta$ are simply the base changes of $\alpha$, $\beta$ to morphisms over $X'$. Hence the lemma follows from Lemma \ref{lemma-ff-base-change-faithful}.
Consider the morphism of schemes $$ g : X = \coprod\nolimits_{i \in I} U_i \longrightarrow Y = \coprod\nolimits_{j \in J} V_j $$ over $S$ which on the $i$th component maps into the $\alpha(i)$th component via the morphism $g_{\alpha(i)}$. We claim that $\{g : X \to Y\}$ is an fpqc covering of schemes. Namely, by Topologies, Lemma \ref{topologies-lemma-disjoint-union-is-fpqc-covering} for each $j$ the morphism $\{\coprod_{\alpha(i) = j} U_i \to V_j\}$ is an fpqc covering. Thus for every affine open $V \subset V_j$ (which we may think of as an affine open of $Y$) we can find finitely many affine opens $W_1, \ldots, W_n \subset \coprod_{\alpha(i) = j} U_i$ (which we may think of as affine opens of $X$) such that $V = \bigcup_{i = 1, \ldots, n} g(W_i)$. This provides enough affine opens of $Y$ which can be covered by finitely many affine opens of $X$ so that Topologies, Lemma \ref{topologies-lemma-recognize-fpqc-covering} part (3) applies, and the claim follows. Let us write $DD(X/S)$, resp.\ $DD(\mathcal{U})$ for the category of descent data with respect to $X/S$, resp.\ $\mathcal{U}$, and similarly for $Y/S$ and $\mathcal{V}$. Consider the diagram $$ \xymatrix{ DD(Y/S) \ar[r] & DD(X/S) \\ DD(\mathcal{V}) \ar[u]^{\text{Lemma }\ref{lemma-family-is-one}} \ar[r] & DD(\mathcal{U}) \ar[u]_{\text{Lemma }\ref{lemma-family-is-one}} } $$ This diagram is commutative, see the proof of Lemma \ref{lemma-pullback-family}. The vertical arrows are equivalences. Hence the lemma follows from Lemma \ref{lemma-fully-faithful} which shows the top horizontal arrow of the diagram is fully faithful.
The functor is faithful and fully faithful by Lemma \ref{lemma-fpqc-refinement-coverings-fully-faithful}. Let us indicate how to prove that it is essentially surjective. Let $(X_i, \varphi_{ii'})$ be a descent datum relative to $\mathcal{U}$. Fix $j \in J$ and set $I_j = \{i \in I \mid \alpha(i) = j\}$. For $i, i' \in I_j$ note that there is a canonical morphism $$ c_{ii'} : U_i \times_{g_i, V_j, g_{i'}} U_{i'} \to U_i \times_S U_{i'}. $$ Hence we can pullback $\varphi_{ii'}$ by this morphism and set $\psi_{ii'} = c_{ii'}^*\varphi_{ii'}$ for $i, i' \in I_j$. In this way we obtain a descent datum $(X_i, \psi_{ii'})$ relative to the Zariski covering $\{g_i : U_i \to V_j\}_{i \in I_j}$. Note that $\psi_{ii'}$ is an isomorphism from the open $X_{i, U_i \times_{V_j} U_{i'}}$ of $X_i$ to the corresponding open of $X_{i'}$. It follows from Schemes, Section \ref{schemes-section-glueing-schemes} that we may glue $(X_i, \psi_{ii'})$ into a scheme $Y_j$ over $V_j$. Moreover, the morphisms $\varphi_{ii'}$ for $i \in I_j$ and $i' \in I_{j'}$ glue to a morphism $\varphi_{jj'} : Y_j \times_S V_{j'} \to V_j \times_S Y_{j'}$ satisfying the cocycle condition (details omitted). Hence we obtain the desired descent datum $(Y_j, \varphi_{jj'})$ relative to $\mathcal{V}$.
Consider the fpqc-covering $\mathcal{W} = \{U_i \times_S V_j \to S\}_{(i, j) \in I \times J}$ of $S$. It is a refinement of both $\mathcal{U}$ and $\mathcal{V}$. Hence we have a $2$-commutative diagram of functors and categories $$ \xymatrix{ DD(\mathcal{V}) \ar[rd] \ar[rr] & & DD(\mathcal{U}) \ar[ld] \\ & DD(\mathcal{W}) & } $$ Notation as in the proof of Lemma \ref{lemma-fpqc-refinement-coverings-fully-faithful} and commutativity by Lemma \ref{lemma-pullback-family} part (3). Hence clearly it suffices to prove the functors $DD(\mathcal{V}) \to DD(\mathcal{W})$ and $DD(\mathcal{U}) \to DD(\mathcal{W})$ are fully faithful. This follows from Lemma \ref{lemma-fpqc-refinement-coverings-fully-faithful} as desired.
Let $V = \bigcup W_i$ be an open covering with $\varphi(W_i \times_S X) \subset X \times_S W_i$ and such that the descent datum $(W_i, \varphi|_{W_i \times_S X})$ is effective. Let $U_i \to S$ be a scheme and let $\alpha_i : (X \times_S U_i, can) \to (W_i, \varphi|_{W_i \times_S X})$ be an isomorphism of descent data. For each pair of indices $(i, j)$ consider the open $\alpha_i^{-1}(W_i \cap W_j) \subset X \times_S U_i$. Because everything is compatible with descent data and since $\{X \to S\}$ is an fpqc covering, we may apply Lemma \ref{lemma-open-fpqc-covering} to find an open $V_{ij} \subset V_j$ such that $\alpha_i^{-1}(W_i \cap W_j) = X \times_S V_{ij}$. Now the identity morphism on $W_i \cap W_j$ is compatible with descent data, hence comes from a unique morphism $\varphi_{ij} : U_{ij} \to U_{ji}$ over $S$ (see Remark \ref{remark-morphisms-of-schemes-satisfy-fpqc-descent}). Then $(U_i, U_{ij}, \varphi_{ij})$ is a glueing data as in Schemes, Section \ref{schemes-section-glueing-schemes} (proof omitted). Thus we may assume there is a scheme $U$ over $S$ such that $U_i \subset U$ is open, $U_{ij} = U_i \cap U_j$ and $\varphi_{ij} = \text{id}_{U_i \cap U_j}$, see Schemes, Lemma \ref{schemes-lemma-glue}. Pulling back to $X$ we can use the $\alpha_i$ to get the desired isomorphism $\alpha : X \times_S U \to V$.