hcju/stacksqa huggingface dataset:
By Lemma \ref{lemma-split-surjection} and Remark \ref{remark-functorial-splitting}, the map $C(1_N \otimes f): C(N \otimes_R S) \to C(N)$ can be split functorially  in $N$. This gives the upper vertical arrows in the commutative diagram $$ \xymatrix@C=8pc{ C(M \otimes_{S, \delta_1^1} S_2) \ar@<1ex>^{C(\theta \circ (1_M \otimes \delta_0^1))}[r] \ar@<-1ex>_{C(1_M \otimes \delta_1^1)}[r] \ar[d] & C(M) \ar[r]\ar[d] & C(f_*(M,\theta)) \ar@{-->}[d] \\ C(M \otimes_{S,\delta_{12}^1} S_3) \ar@<1ex>^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))}[r] \ar@<-1ex>_{C(1_{M \otimes S_2} \otimes \delta^2_1)}[r] \ar[d] & C(M \otimes_{S, \delta_1^1} S_2 ) \ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} \ar[d]^{C(1_M \otimes \delta_1^1)} & C(M) \ar[d] \ar@{=}[dl] \\ C(M \otimes_{S, \delta_1^1} S_2) \ar@<1ex>[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} \ar@<-1ex>[r]_{C(1_M \otimes \delta_1^1)} & C(M) \ar[r] & C(f_*(M,\theta)) } $$ in which the compositions along the columns are identity morphisms. The second row is the coequalizer diagram (\ref{equation-coequalizer-CM}); this produces the dashed arrow. From the top right square, we obtain auxiliary morphisms $C(f_*(M,\theta)) \to  C(M)$  and $C(M) \to C(M\otimes_{S,\delta_1^1} S_2)$ which imply that the first row is  a split coequalizer diagram. By Remark \ref{remark-adjunction}, we may tensor with $S$ inside $C$ to obtain  the split coequalizer diagram $$ \xymatrix@C=8pc{ C(M \otimes_{S,\delta_2^2 \circ \delta_1^1} S_3) \ar@<1ex>^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))}[r]  \ar@<-1ex>_{C(1_{M \otimes S_2} \otimes \delta^2_1)}[r] & C(M \otimes_{S, \delta_1^1} S_2 ) \ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} & C(f_*(M,\theta) \otimes_R S). } $$ By Lemma \ref{lemma-C-is-faithful}, we conclude (\ref{equation-equalizer-f2}) must also be an equalizer.
Let $0 \to N \to P \to Q \to 0$ be an exact sequence in $\text{Mod}_R$. Since  $M$ is flat, $$ 0 \to N \otimes_R M \to P \otimes_R M \to Q \otimes_R M \to 0 $$ is exact. By Lemma \ref{lemma-C-is-faithful}, $$ 0 \to C(Q \otimes_R M) \to C(P \otimes_R M) \to C(N \otimes_R M) \to 0 $$ is exact. By (\ref{equation-adjunction}), this last sequence can be rewritten as $$ 0 \to \Hom_R(Q, C(M)) \to \Hom_R(P, C(M)) \to \Hom_R(N, C(M)) \to 0. $$ Hence $C(M)$ is an injective object of $\text{Mod}_R$.
This is a restatement of Proposition \ref{proposition-descent-module} in terms of schemes. First, note that a descent datum $\xi$ for quasi-coherent sheaves with respect to $\mathcal{U}$ is exactly the same as a descent datum $\xi'$ for quasi-coherent sheaves with respect to the covering $\mathcal{U}' = \{\coprod_{i = 1, \ldots, n} U_i \to S\}$. Moreover, effectivity for $\xi$ is the same as effectivity for $\xi'$. Hence we may assume $n = 1$, i.e., $\mathcal{U} = \{U \to S\}$ where $U$ and $S$ are affine. In this case descent data correspond to descent data on modules with respect to the ring map $$ \Gamma(S, \mathcal{O}) \longrightarrow \Gamma(U, \mathcal{O}). $$ Since $U \to S$ is surjective and flat, we see that this ring map is faithfully flat. In other words, Proposition \ref{proposition-descent-module} applies and we win.
Since $X \to Y$ is surjective finite \'etale $\{X \to Y\}$ is an fpqc covering. Since $G \times X \to X \times_Y X$, $(\sigma, x) \mapsto (x, f_\sigma(x))$ is an isomorphism, we see that $G \times G \times X \to X \times_Y X \times_Y X$, $(\sigma, \tau, x) \mapsto (x, f_\sigma(x), f_{\sigma\tau}(x))$ is an isomorphism too. Using these identifications, the category of data as in the lemma is the same as the category of descent data for quasi-coherent sheaves for the covering $\{x \to Y\}$. Thus the lemma follows from Proposition \ref{proposition-fpqc-descent-quasi-coherent}.
Let $y \in Y$ be a point. Choose a point $x \in X$ mapping to $y$. Choose an open $x \in U \subset X$ and elements $s_1, \ldots, s_n$ of $f^*\mathcal{F}(U)$ which generate $f^*\mathcal{F}$ over $U$. Since $f^*\mathcal{F} = f^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X$ we can after shrinking $U$ assume $s_i = \sum t_{ij} \otimes a_{ij}$ with $t_{ij} \in f^{-1}\mathcal{F}(U)$ and $a_{ij} \in \mathcal{O}_X(U)$. After shrinking $U$ further we may assume that $t_{ij}$ comes from a section $s_{ij} \in \mathcal{F}(V)$ for some $V \subset Y$ open with $f(U) \subset V$. Let $N$ be the number of sections $s_{ij}$ and consider the map $$ \sigma = (s_{ij}) : \mathcal{O}_V^{\oplus N} \to \mathcal{F}|_V $$ By our choice of the sections we see that $f^*\sigma|_U$ is surjective. Hence for every $u \in U$ the map $$ \sigma_{f(u)} \otimes_{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u} : \mathcal{O}_{X, u}^{\oplus N} \longrightarrow \mathcal{F}_{f(u)} \otimes_{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u} $$ is surjective. As $f$ is flat, the local ring map $\mathcal{O}_{Y, f(u)} \to \mathcal{O}_{X, u}$ is flat, hence faithfully flat (Algebra, Lemma \ref{algebra-lemma-local-flat-ff}). Hence $\sigma_{f(u)}$ is surjective. Since $f$ is open, $f(U)$ is an open neighbourhood of $y$ and the proof is done.
By Lemma \ref{lemma-finite-type-descends} we see that $\mathcal{F}$ is of finite type. Hence Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) implies that $\mathcal{F}$ is generated by $r$ sections in the neighbourhood of a point $x \in X$ if and only if $\dim_{\kappa(x)} \mathcal{F}_x \otimes \kappa(x) \leq r$. Choose an $i$ and a point $x_i \in X_i$ mapping to $x$. Then $\dim_{\kappa(x)} \mathcal{F}_x \otimes \kappa(x) =  \dim_{\kappa(x_i)} (f_i^*\mathcal{F})_{x_i} \otimes \kappa(x_i)$ which is $\leq r$ as $f_i^*\mathcal{F}$ is locally generated by $r$ sections.
This follows from the fact that a quasi-coherent sheaf is finite locally free if and only if it is of finite presentation and flat, see Algebra, Lemma \ref{algebra-lemma-finite-projective}. Namely, if each $f_i^*\mathcal{F}$ is flat and of finite presentation, then so is $\mathcal{F}$ by Lemmas \ref{lemma-flat-descends} and \ref{lemma-finite-presentation-descends}.
As $f$ is finite it is affine. This reduces us to the case where $f$ is the morphism $\Spec(B) \to \Spec(A)$ given by a finite ring map $A \to B$. Moreover, then $\mathcal{F} = \widetilde{M}$ is the sheaf of modules associated to the $B$-module $M$. Note that $M$ is finite as a $B$-module if and only if $M$ is finite as an $A$-module, see Algebra, Lemma \ref{algebra-lemma-finite-module-over-finite-extension}. Combined with Properties, Lemma \ref{properties-lemma-finite-type-module} this proves the lemma.
As $f$ is finite it is affine. This reduces us to the case where $f$ is the morphism $\Spec(B) \to \Spec(A)$ given by a finite and finitely presented ring map $A \to B$. Moreover, then $\mathcal{F} = \widetilde{M}$ is the sheaf of modules associated to the $B$-module $M$. Note that $M$ is finitely presented as a $B$-module if and only if $M$ is finitely presented as an $A$-module, see Algebra, Lemma \ref{algebra-lemma-finite-finitely-presented-extension}. Combined with Properties, Lemma \ref{properties-lemma-finite-presentation-module} this proves the lemma.
For $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$ a $\tau$-covering is also a fpqc-covering, see the results in Topologies, Lemmas \ref{topologies-lemma-zariski-etale}, \ref{topologies-lemma-zariski-etale-smooth}, \ref{topologies-lemma-zariski-etale-smooth-syntomic}, \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf}, and \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}. Hence it suffices to prove the theorem for a fpqc covering. Assume that $\{f_i : T_i \to T\}_{i \in I}$ is an fpqc covering where $f : T \to S$ is given. Suppose that we have a family of sections $s_i \in \Gamma(T_i , f_i^*f^*\mathcal{F})$ such that $s_i|_{T_i \times_T T_j} = s_j|_{T_i \times_T T_j}$. We have to find the correspond section $s \in \Gamma(T, f^*\mathcal{F})$. We can reinterpret the $s_i$ as a family of maps $\varphi_i : f_i^*\mathcal{O}_T = \mathcal{O}_{T_i} \to f_i^*f^*\mathcal{F}$ compatible with the canonical descent data associated to the quasi-coherent sheaves $\mathcal{O}_T$ and $f^*\mathcal{F}$ on $T$. Hence by Proposition \ref{proposition-fpqc-descent-quasi-coherent} we see that we may (uniquely) descend these to a map $\mathcal{O}_T \to f^*\mathcal{F}$ which gives us our section $s$.
Proof of (1). We first note that the result is true when $\tau = Zar$ because in that case we have the morphism of topoi $i_f : \Sh(T_{Zar}) \to \Sh((\Sch/S)_{Zar})$ such that $\text{id}_{\tau, Zar} \circ i_f = f_{small}$ as morphisms $T_{Zar} \to S_{Zar}$, see Topologies, Lemmas \ref{topologies-lemma-put-in-T} and \ref{topologies-lemma-morphism-big-small}. Since pullback is transitive (see Modules on Sites, Lemma \ref{sites-modules-lemma-push-pull-composition-modules}) we see that $i_f^*(\text{id}_{\tau, Zar})^*\mathcal{F} = f_{small}^*\mathcal{F}$ as desired. Hence, by the remark preceding this lemma we see that $(\text{id}_{\tau, Zar})^*\mathcal{F}$ is the $\tau$-sheafification of the presheaf $T \mapsto \Gamma(T, f^*\mathcal{F})$. \medskip\noindent The proof of (3) is exactly the same as the proof of (1), except that it uses Topologies, Lemmas \ref{topologies-lemma-put-in-T-etale} and \ref{topologies-lemma-morphism-big-small-etale}. We omit the proof of (2).
Let $\{S_i \to S\}$ be a Zariski covering such that we have exact sequences $$ \bigoplus\nolimits_{k \in K_i} \mathcal{O}_{S_i} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}_{S_i} \longrightarrow \mathcal{F} \longrightarrow 0 $$ for some index sets $K_i$ and $J_i$. This is possible by the definition of a quasi-coherent sheaf on a ringed space (See Modules, Definition \ref{modules-definition-quasi-coherent}). \medskip\noindent Proof of (1). Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. It is clear that $\mathcal{F}^a|_{(\Sch/S_i)_\tau}$ also sits in an exact sequence $$ \bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \mathcal{F}^a|_{(\Sch/S_i)_\tau} \longrightarrow 0 $$ Hence $\mathcal{F}^a$ is quasi-coherent by Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}. \medskip\noindent Proof of (2). Let $\tau = \etale$. It is clear that $\mathcal{F}^a|_{(S_i)_\etale}$ also sits in an exact sequence $$ \bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(S_i)_\etale} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(S_i)_\etale} \longrightarrow \mathcal{F}^a|_{(S_i)_\etale} \longrightarrow 0 $$ Hence $\mathcal{F}^a$ is quasi-coherent by Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}. The case $\tau = Zariski$ is similar (actually, it is really tautological since the corresponding ringed topoi agree).
By Lemma \ref{lemma-standard-covering-Cech} we see that $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a)$ is isomorphic to $\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}^a)$ where $\mathcal{V} = \{V \to U\}$ with $V = \coprod_{i = 1, \ldots n} U_i$ affine also. Set $A = \Gamma(V, \mathcal{O}_V)$. Since $\{V \to U\}$ is a $\tau$-covering we see that $R \to A$ is faithfully flat. On the other hand, by definition of $\mathcal{F}^a$ we have that the degree $p$ term $\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{F}^a)$ is $$ \Gamma(V \times_U \ldots \times_U V, \mathcal{F}^a) = \Gamma(\Spec(A \otimes_R \ldots \otimes_R A), \mathcal{F}^a) = A \otimes_R \ldots \otimes_R A \otimes_R M $$ We omit the verification that the maps of the {\v C}ech complex agree with the maps in the complex $s((A/R)_\bullet \otimes_R M)$. The vanishing of cohomology is Lemma \ref{lemma-ff-exact}.
This is immediate for the global properties, i.e., those defined in Modules on Sites, Definition \ref{sites-modules-definition-global}. For the local properties we can use Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object} to translate ``$\mathcal{F}^a$ has $\mathcal{P}$'' into a property on the members of a covering of $X$. Hence the result follows from Lemmas \ref{lemma-finite-type-descends}, \ref{lemma-finite-presentation-descends}, \ref{lemma-locally-generated-by-r-sections-descends}, \ref{lemma-flat-descends}, and \ref{lemma-finite-locally-free-descends}. Being coherent for a quasi-coherent module is the same as being of finite type over a locally Noetherian scheme (see Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}) hence this reduces to the case of finite type modules (details omitted).
We prove this for the \'etale topology; we omit the proof in the case of the Zariski topology. By Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images} the sheaves $R^if_*\mathcal{F}$ are quasi-coherent so that the assertion makes sense. The sheaf $R^if_{small, *}\mathcal{F}^a$ is the sheaf associated to the presheaf $$ U \longmapsto H^i(U \times_S T, \mathcal{F}^a) $$ where $g : U \to S$ is an object of $S_\etale$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. By our conventions the right hand side is the \'etale cohomology of the restriction of $\mathcal{F}^a$ to the localization $T_\etale/U \times_S T$ which equals $(U \times_S T)_\etale$. By Proposition \ref{proposition-same-cohomology-quasi-coherent} this is presheaf the same as the presheaf $$ U \longmapsto H^i(U \times_S T, (g')^*\mathcal{F}), $$ where $g' : U \times_S T \to T$ is the projection. If $U$ is affine then this is the same as $H^0(U, R^if'_*(g')^*\mathcal{F})$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images-application}. By Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology} this is equal to $H^0(U, g^*R^if_*\mathcal{F})$ which is the value of $(R^if_*\mathcal{F})^a$ on $U$. Thus the values of the sheaves of modules $R^if_{small, *}(\mathcal{F}^a)$ and $(R^if_*\mathcal{F})^a$ on every affine object of $S_\etale$ are canonically isomorphic which implies they are canonically isomorphic.
Proof in case $\tau = fppf$; the other cases are proved in the exact same way. The assumption means that $\mathcal{G}(U) = 0$ for any $U \to S$ flat and locally of finite presentation. Apply Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection} to the subset $\mathcal{B} \subset \Ob((\Sch/S)_{fppf})$ consisting of $U \to S$ flat and locally of finite presentation and the collection $\text{Cov}$ of all fppf coverings of elements of $\mathcal{B}$.
Recall that $R^if_*\mathcal{F}$ is the sheaf associated to the presheaf $$ U \mapsto H^i((\Sch/U \times_S T)_\tau, \mathcal{F}) $$ see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. If $U \to S$ is flat, then $U \times_S T \to T$ is flat as a base change. Hence the displayed group is zero by Lemma \ref{lemma-cohomology-parasitic}. If $\{U_i \to U\}$ is a $\tau$-covering then $U_i \times_S T \to T$ is also flat. Hence it is clear that the sheafification of the displayed presheaf is zero on schemes $U$ flat over $S$.
Let $\mathcal{F}$ be the pullback of $\mathcal{G}|_{S_\tau}$ to the big fppf site $(\Sch/S)_{fppf}$. Note that $\mathcal{F}$ is quasi-coherent. There is a canonical comparison map $\varphi : \mathcal{F} \to \mathcal{G}$ which by assumptions (1) and (2) induces an isomorphism $\mathcal{F}|_{U_\tau} \to \mathcal{G}|_{U_\tau}$ for all $g : U \to S$ flat and locally of finite presentation. Hence in the short exact sequences $$ 0 \to \Ker(\varphi) \to \mathcal{F} \to \Im(\varphi) \to 0 $$ and $$ 0 \to \Im(\varphi) \to \mathcal{G} \to \Coker(\varphi) \to 0 $$ the sheaves $\Ker(\varphi)$ and $\Coker(\varphi)$ are parasitic for the fppf topology. By Lemma \ref{lemma-cohomology-parasitic} we conclude that $H^p(U, \mathcal{F}) \to H^p(U, \mathcal{G})$ is an isomorphism for $g : U \to S$ flat and locally of finite presentation. Since the result holds for $\mathcal{F}$ by Proposition \ref{proposition-same-cohomology-quasi-coherent} we win.
Assume (2) holds and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider the scheme (Constructions, Section \ref{constructions-section-spec}) $$ X' = \underline{\Spec}_X(\mathcal{O}_X \oplus \mathcal{F}) $$ where $\mathcal{O}_X \oplus \mathcal{F}$ is an $\mathcal{O}_X$-algebra with multiplication $(f, s)(f', s') = (ff', fs' + f's)$. If $s_i \in \Gamma(X_i, f_i^*\mathcal{F})$ is a section, then $s_i$ determines a unique element of $$ \Gamma(X' \times_X X_i, \mathcal{O}_{X' \times_X X_i}) = \Gamma(X_i, \mathcal{O}_{X_i}) \oplus \Gamma(X_i, f_i^*\mathcal{F}) $$ Proof of equality omitted. If $(s_i)_{i \in I}$ is in the equalizer of (1), then, using the equality $$ \Mor(T, \mathbf{A}^1_\mathbf{Z}) = \Gamma(T, \mathcal{O}_T) $$ which holds for any scheme $T$, we see that these sections define a family of morphisms $h_i : X' \times_X X_i \to \mathbf{A}^1_\mathbf{Z}$ with $h_i \circ \text{pr}_1 = h_j \circ \text{pr}_2$ as morphisms $(X' \times_X X_i) \times_{X'} (X' \times_X X_j) \to \mathbf{A}^1_\mathbf{Z}$. Since we've assume (2) we obtain a morphism $h : X' \to \mathbf{A}^1_\mathbf{Z}$ compatible with the morphisms $h_i$ which in turn determines an element $s \in \Gamma(X, \mathcal{F})$. We omit the verification that $s$ maps to $s_i$ in $\Gamma(X_i, f_i^*\mathcal{F})$. \medskip\noindent Assume (1). Let $T$ be an affine scheme and let $h_i : X_i \to T$ be a family of morphisms such that $h_i \circ \text{pr}_1 = h_j \circ \text{pr}_2$ on $X_i \times_X X_j$ for all $i, j \in I$. Then $$ \prod h_i^\sharp : \Gamma(T, \mathcal{O}_T) \to \prod \Gamma(X_i, \mathcal{O}_{X_i}) $$ maps into the equalizer and we find that we get a ring map $\Gamma(T, \mathcal{O}_T) \to \Gamma(X, \mathcal{O}_X)$ by the assumption of the lemma for $\mathcal{F} = \mathcal{O}_X$. This ring map corresponds to a morphism $h : X \to T$ such that $h_i = h \circ f_i$. Hence our family is an effective epimorphism. \medskip\noindent Let $p : Y \to X$ be a morphism of affines. We will show the base changes $g_i : Y_i \to Y$ of $f_i$ form an effective epimorphism by applying the result of the previous paragraph. Namely, if $\mathcal{G}$ is a quasi-coherent $\mathcal{O}_Y$-module, then $$ \Gamma(Y, \mathcal{G}) = \Gamma(X, p_*\mathcal{G}),\quad \Gamma(Y_i, g_i^*\mathcal{G}) = \Gamma(X, f_i^*p_*\mathcal{G}), $$ and $$ \Gamma(Y_i \times_Y Y_j, (g_i \times g_j)^*\mathcal{G}) = \Gamma(X, (f_i \times f_j)^*p_*\mathcal{G}) $$ by the trivial base change formula (Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-base-change}). Thus we see property (1) lemma holds for the family $g_i$.
By Lemma \ref{lemma-check-universal-effective-epimorphism-affine} it suffices to base change our family of morphisms by $Y \to X$ with $Y$ affine. Set $Y_i = X_i \times_X Y$. Let $T$ be a scheme and let $h_i : Y_i \to Y$ be a family of morphisms such that $h_i \circ \text{pr}_1 = h_j \circ \text{pr}_2$ on $Y_i \times_Y Y_j$. Note that $Y$ as a set is the coequalizer of the two maps from $\coprod Y_i \times_Y Y_j$ to $\coprod Y_i$. Namely, surjectivity by the affine case of Lemma \ref{lemma-universal-effective-epimorphism-surjective} and injectivity by Lemma \ref{lemma-equiv-fibre-product}. Hence there is a set map of underlying sets $h : Y \to T$ compatible with the maps $h_i$. By the second condition of the lemma we see that $h$ is continuous! Thus if $y \in Y$ and $U \subset T$ is an affine open neighbourhood of $h(y)$, then we can find an affine open $V \subset Y$ such that $h(V) \subset U$. Setting $V_i = Y_i \times_Y V = X_i \times_X V$ we can use the result proved in Lemma \ref{lemma-universal-effective-epimorphism-affine} to see that $h|_V : V \to U \subset T$ comes from a unique morphism of affine schemes $h_V : V \to U$ agreeing with $h_i|_{V_i}$ as morphisms of schemes for all $i$. Glueing these $h_V$ (see Schemes, Section \ref{schemes-section-glueing-schemes}) gives a morphism $Y \to T$ as desired.
Apply Lemma \ref{lemma-equiv-fibre-product} to the map $\coprod_{i \in I} T_i \to T$. It implies there exists a subset $W \subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$, namely $W = \bigcup f_i(W_i)$. To see that $W$ is open we may work Zariski locally on $T$. Hence we may assume that $T$ is affine. Using the definition of a fpqc covering, this reduces us to the case where $\{f_i : T_i \to T\}$ is a standard fpqc covering. In this case we may apply Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology} to the morphism $\coprod T_i \to T$ to conclude that $W$ is open.
Let $S$ be a scheme. We have to show the following: Given morphisms $\varphi_i : T_i \to S$ such that $\varphi_i|_{T_i \times_T T_j} = \varphi_j|_{T_i \times_T T_j}$ there exists a unique morphism $T \to S$ which restricts to $\varphi_i$ on each $T_i$. In other words, we have to show that the functor $h_S = \Mor_{\Sch}( - , S)$ satisfies the sheaf property for the fpqc topology. \medskip\noindent If $\{T_i \to T\}$ is a Zariski covering, then this follows from Schemes, Lemma \ref{schemes-lemma-glue}. Thus Topologies, Lemma \ref{topologies-lemma-sheaf-property-fpqc} reduces us to the case of a covering $\{X \to Y\}$ given by a single surjective flat morphism of affines. \medskip\noindent First proof. By Lemma \ref{lemma-sheaf-condition-holds} we have the sheaf condition for quasi-coherent modules for $\{X \to Y\}$. By Lemma \ref{lemma-open-fpqc-covering} the morphism $X \to Y$ is universally submersive. Hence we may apply Lemma \ref{lemma-universal-effective-epimorphism} to see that $\{X \to Y\}$ is a universal effective epimorphism. \medskip\noindent Second proof. Let $R \to A$ be the faithfully flat ring map corresponding to our surjective flat morphism $\pi : X \to Y$. Let $f : X \to S$ be a morphism such that $f \circ \text{pr}_1 = f \circ \text{pr}_2$ as morphisms $X \times_Y X = \Spec(A \otimes_R A) \to S$. By Lemma \ref{lemma-equiv-fibre-product} we see that as a map on the underlying sets $f$ is of the form $f = g \circ \pi$ for some (set theoretic) map $g : \Spec(R) \to S$. By Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology} and the fact that $f$ is continuous we see that $g$ is continuous. \medskip\noindent Pick $y \in Y = \Spec(R)$. Choose $U \subset S$ affine open containing $g(y)$. Say $U = \Spec(B)$. By the above we may choose an $r \in R$ such that $y \in D(r) \subset g^{-1}(U)$. The restriction of $f$ to $\pi^{-1}(D(r))$ into $U$ corresponds to a ring map $B \to A_r$. The two induced ring maps $B \to A_r \otimes_{R_r} A_r = (A \otimes_R A)_r$ are equal by assumption on $f$. Note that $R_r \to A_r$ is faithfully flat. By Lemma \ref{lemma-ff-exact} the equalizer of the two arrows $A_r \to A_r \otimes_{R_r} A_r$ is $R_r$. We conclude that $B \to A_r$ factors uniquely through a map $B \to R_r$. This map in turn gives a morphism of schemes $D(r) \to U \to S$, see Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}. \medskip\noindent What have we proved so far? We have shown that for any prime $\mathfrak p \subset R$, there exists a standard affine open $D(r) \subset \Spec(R)$ such that the morphism $f|_{\pi^{-1}(D(r))} : \pi^{-1}(D(r)) \to S$ factors uniquely through some morphism of schemes $D(r) \to S$. We omit the verification that these morphisms glue to the desired morphism $\Spec(R) \to S$.
By Algebra, Lemma \ref{algebra-lemma-descend-faithfully-flat-finite-presentation} there exists a commutative diagram $$ \xymatrix{ R \ar[r] \ar@{=}[d] & A_0 \ar[d] \ar[r] & B_0 \ar[d] \\ R \ar[r] & A \ar[r] & B } $$ with $R \to A_0$ of finite presentation, $A_0 \to B_0$ faithfully flat of finite presentation and $B = A \otimes_{A_0} B_0$. Since $R \to B$ is of finite type by assumption, we may add some elements to $A_0$ and assume that the map $B_0 \to B$ is surjective! In this case, since $A_0 \to B_0$ is faithfully flat, we see that as $$ (A_0 \to A) \otimes_{A_0} B_0 \cong (B_0 \to B) $$ is surjective, also $A_0 \to A$ is surjective. Hence we win.