hcju/stacksqa huggingface dataset:
Part (1) follows from the fact that any Zariski open covering of $T$ can be refined by a finite affine open covering. \medskip\noindent Proof of (3). Choose $U \to T$ proper surjective and $U = \bigcup_{j = 1, \ldots, m} U_j$ as in Definition \ref{definition-standard-ph-covering}. Choose $W_j \to U_j$ proper surjective and $W_j = \bigcup W_{ji}$ as in Definition \ref{definition-standard-ph-covering}. By Chow's lemma (Limits, Lemma \ref{limits-lemma-chow-finite-type}) we can find $W'_j \to W_j$ proper surjective and closed immersions $W'_j \to \mathbf{P}^{e_j}_{U_j}$. Thus, after replacing $W_j$ by $W'_j$ and $W_j = \bigcup W_{ji}$ by a suitable affine open covering of $W'_j$, we may assume there is a closed immersion $W_j \subset \mathbf{P}^{e_j}_{U_j}$ for all $j = 1, \ldots, m$. \medskip\noindent Let $\overline{W}_j \subset \mathbf{P}^{e_j}_U$ be the scheme theoretic closure of $W_j$. Then $W_j \subset \overline{W}_j$ is an open subscheme; in fact $W_j$ is the inverse image of $U_j \subset U$ under the morphism $\overline{W}_j \to U$. (To see this use that $W_j \to \mathbf{P}^{e_j}_U$ is quasi-compact and hence formation of the scheme theoretic image commutes with restriction to opens, see Morphisms, Section \ref{morphisms-section-scheme-theoretic-image}.) Let $Z_j = U \setminus U_j$ with reduced induced closed subscheme structure. Then $$ V_j = \overline{W}_j \amalg Z_j \to U $$ is proper surjective and the open subscheme $W_j \subset V_j$ is the inverse image of $U_j$. Hence for $v \in V_j$, $v \not \in W_j$ we can pick an affine open neighbourhood $v \in V_{j, v} \subset V_j$ which maps into $U_{j'}$ for some $1 \leq j' \leq m$. \medskip\noindent To finish the proof we consider the proper surjective morphism $$ V = V_1 \times_U V_2 \times_U \ldots \times_U V_m \longrightarrow U \longrightarrow T $$ and the covering of $V$ by the affine opens $$ V_{1, v_1} \times_U \ldots \times_U V_{j - 1, v_{j - 1}} \times_U W_{j i} \times_U V_{j + 1, v_{j + 1}} \times_U \ldots \times_U V_{m, v_m} $$ These do indeed form a covering, because each point of $U$ is in some $U_j$ and the inverse image of $U_j$ in $V$ is equal to $V_1 \times \ldots \times V_{j - 1} \times W_j \times V_{j + 1} \times \ldots \times V_m$. Observe that the morphism from the affine open displayed above to $T$ factors through $W_{ji}$ thus we obtain a refinement. Finally, we only need a finite number of these affine opens as $V$ is quasi-compact (as a scheme proper over the affine scheme $T$).
The equivalence of (1) and (2) follows immediately from Definition \ref{definition-ph-covering} and the fact that a refinement of a refinement is a refinement. Because of the equivalence of (1) and (2) and since $\{T_i \to T\}_{i \in I}$ refines $\{\coprod_{i \in I} T_i \to T\}$ we see that (1) implies (3). Finally, assume (3) holds. Let $U \subset T$ be an affine open and let $\{U_j \to U\}_{j = 1, \ldots, m}$ be a standard ph covering which refines $\{U \times_T \coprod_{i \in I} T_i \to U\}$. This means that for each $j$ we have a morphism $$ h_j : U_j \longrightarrow U \times_T \coprod\nolimits_{i \in I} T_i = \coprod\nolimits_{i \in I} U \times_T T_i $$ over $U$. Since $U_j$ is quasi-compact, we get disjoint union decompositions $U_j = \coprod_{i \in I} U_{j, i}$ by open and closed subschemes almost all of which are empty such that $h_j|_{U_{j, i}}$ maps $U_{j, i}$ into $U \times_T T_i$. It follows that $$ \{U_{j, i} \to U\}_{j = 1, \ldots, m,\ i \in I,\ U_{j, i} \not = \emptyset} $$ is a standard ph covering (small detail omitted) refining $\{U \times_T T_i \to U\}_{i \in I}$. Thus (1) holds.
Assertion (1) is clear. \medskip\noindent Proof of (3). The base change $T_i \times_T T' \to T'$ is locally of finite type by Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-type}. hence we only need to check the condition on affine opens. Let $U' \subset T'$ be an affine open subscheme. Since $U'$ is quasi-compact we can find a finite affine open covering $U' = U'_1 \cup \ldots \cup U'$ such that $U'_j \to T$ maps into an affine open $U_j \subset T$. Choose a standard ph covering $\{U_{jl} \to U_j\}_{l = 1, \ldots, n_j}$ refining $\{T_i \times_T U_j \to U_j\}$. By Lemma \ref{lemma-base-change-standard-ph} the base change $\{U_{jl} \times_{U_j} U'_j \to U'_j\}$ is a standard ph covering. Note that $\{U'_j \to U'\}$ is a standard ph covering as well. By Lemma \ref{lemma-refine-by-standard-ph}  the family $\{U_{jl} \times_{U_j} U'_j \to U'\}$ can be refined by a standard ph covering. Since $\{U_{jl} \times_{U_j} U'_j \to U'\}$ refines $\{T_i \times_T U' \to U'\}$ we conclude. \medskip\noindent Proof of (2). Composition preserves being locally of finite type, see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-type}. Hence we only need to check the condition on affine opens. Let $U \subset T$ be affine open. First we pick a standard ph covering $\{U_k \to U\}_{k = 1, \ldots, m}$ refining $\{T_i \times_T U \to U\}$. Say the refinement is given by morphisms $U_k \to T_{i_k}$ over $T$. Then $$ \{T_{i_kj} \times_{T_{i_k}} U_k \to U_k\}_{j \in J_{i_k}} $$ is a ph covering by part (3). As $U_k$ is affine, we can find a standard ph covering $\{U_{ka} \to U_k\}_{a = 1, \ldots, b_k}$ refining this family. Then we apply Lemma \ref{lemma-refine-by-standard-ph} to see that $\{U_{ka} \to U\}$ can be refined by a standard ph covering. Since $\{U_{ka} \to U\}$ refines $\{T_{ij} \times_T U \to U\}$ this finishes the proof.
For each $i$ choose an affine open covering $T_i = \bigcup_{j \in J_i} T_{ij}$ such that each $T_{ij}$ maps into an affine open subscheme of $T$. By Lemmas \ref{lemma-zariski-ph} and \ref{lemma-ph} the refinement $\{T_{ij} \to T\}_{i \in I, j \in J_i}$ is a ph covering of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into an affine open $W_i$ of $T$. Applying Sets, Lemma \ref{sets-lemma-what-is-in-it} we see that $W_i$ is isomorphic to an object of $\Sch_{ph}$. But then $T_i$ as a finite type scheme over $W_i$ is isomorphic to an object $V_i$ of $\Sch_{ph}$ by a second application of Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{V_i \to T\}_{i \in I}$ refines $\{T_i \to T\}_{i \in I}$ (because they are isomorphic). Moreover, $\{V_i \to T\}_{i \in I}$ is combinatorially equivalent to a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_{ph}$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{U_j \to T\}_{j \in J}$ is a refinement as in (1). In the situation of (2), (3) each of the schemes $T_i$ is isomorphic to an object of $\Sch_{ph}$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}, and another application of Sets, Lemma \ref{sets-lemma-coverings-site} gives what we want.
Reasoning as in the proof of Lemma \ref{lemma-verify-site-etale} it suffices to show that the collection of finite ph coverings $\{U_i \to U\}$ with $U$, $U_i$ affine satisfies properties (1), (2) and (3) of Sites, Definition \ref{sites-definition-site}. This is clear since for example, given a finite ph covering $\{T_i \to T\}_{i\in I}$ with $T_i, T$ affine, and for each $i$ a finite ph covering $\{T_{ij} \to T_i\}_{j\in J_i}$ with $T_{ij}$ affine , then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a ph covering (Lemma \ref{lemma-ph}), $\bigcup_{i\in I} J_i$ is finite and each $T_{ij}$ is affine.
For $\Sch_{ph}$ it is true by construction, see Sets, Lemma \ref{sets-lemma-what-is-in-it}. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \Ob(\Sch_{ph})$. The fibre product $V \times_U W$ in $\Sch_{ph}$ is a fibre product in $\Sch$ and is the fibre product of $V/S$ with $W/S$ over $U/S$ in the category of all schemes over $S$, and hence also a fibre product in $(\Sch/S)_{ph}$. This proves the result for $(\Sch/S)_{ph}$. If $U, V, W$ are affine, so is $V \times_U W$ and hence the result for $(\textit{Aff}/S)_{ph}$.
The notion of a special cocontinuous functor is introduced in Sites, Definition \ref{sites-definition-special-cocontinuous-functor}. Thus we have to verify assumptions (1) -- (5) of Sites, Lemma \ref{sites-lemma-equivalence}. Denote the inclusion functor $u : (\textit{Aff}/S)_{ph} \to (\Sch/S)_{ph}$. Being cocontinuous follows because any ph covering of $T/S$, $T$ affine, can be refined by a standard ph covering of $T$ by definition. Hence (1) holds. We see $u$ is continuous simply because a finite ph covering of an affine by affines is a ph covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering (which is a ph covering).
Suppose that $U \subset T$ is an affine open. If (1) holds, then we find $i_1, \ldots, i_n \in I$ and affine opens $U_j \subset T_{i_j}$ such that $U = \bigcup_{j = 1, \ldots, n} f_{i_j}(U_j)$. Then $U_1 \amalg \ldots \amalg U_n \subset T'$ is a quasi-compact open surjecting onto $U$. Thus $\{f : T' \to T\}$ is an fpqc covering by Lemma \ref{lemma-recognize-fpqc-covering}. Conversely, if (2) holds then there exists a quasi-compact open $U' \subset T'$ with $U = f(U')$. Then $U_j = U' \cap T_j$ is quasi-compact open in $T_j$ and empty for almost all $j$. By Lemma \ref{lemma-recognize-fpqc-covering} we see that (1) holds.
Let $\{g_j : X_j \to T\}_{j \in J}$ be an fpqc covering refining $\{f_i : T_i \to T\}$. Suppose that $U \subset T$ is affine open. Choose $j_1, \ldots, j_m \in J$ and $V_k \subset X_{j_k}$ affine open such that $U = \bigcup g_{j_k}(V_k)$. For each $j$ pick $i_j \in I$ and a morphism $h_j : X_j \to T_{i_j}$ such that $g_j = f_{i_j} \circ h_j$. Since $h_{j_k}(V_k)$ is quasi-compact we can find a quasi-compact open $h_{j_k}(V_k) \subset U_k \subset f_{i_{j_k}}^{-1}(U)$. Then $U = \bigcup f_{i_{j_k}}(U_k)$. We conclude that $\{f_i : T_i \to T\}_{i \in I}$ is an fpqc covering by Lemma \ref{lemma-recognize-fpqc-covering}.
We will use Lemma \ref{lemma-recognize-fpqc-covering} without further mention. Let $U \subset T$ be an affine open. By (2) we can find quasi-compact opens $V_j \subset S_j$ for $j \in J$, almost all empty, such that $U = \bigcup g_j(V_j)$. Then for each $j$ we can choose quasi-compact opens $W_{ij} \subset S_j \times_T T_i$ for $i \in I$, almost all empty, with $V_j = \bigcup_i \text{pr}_1(W_{ij})$. Thus $\{S_j \times_T T_i \to T\}$ is an fpqc covering. Since this covering refines $\{f_i : T_i \to T\}$ we conclude by Lemma \ref{lemma-family-flat-dominated-covering}.
We will show that an fppf covering is an fpqc covering, and then the rest follows from Lemma \ref{lemma-zariski-etale-smooth-syntomic-fppf}. Let $\{f_i : U_i \to U\}_{i \in I}$ be an fppf covering. By definition this means that the $f_i$ are flat which checks the first condition of Definition \ref{definition-fpqc-covering}. To check the second, let $V \subset U$ be an affine open subset. Write $f_i^{-1}(V) = \bigcup_{j \in J_i} V_{ij}$ for some affine opens $V_{ij} \subset U_i$. Since each $f_i$ is open (Morphisms, Lemma \ref{morphisms-lemma-fppf-open}), we see that $V = \bigcup_{i\in I} \bigcup_{j \in J_i} f_i(V_{ij})$ is an open covering of $V$. Since $V$ is quasi-compact, this covering has a finite refinement. This finishes the proof.
Part (1) is immediate. Recall that the composition of flat morphisms is flat and that the base change of a flat morphism is flat (Morphisms, Lemmas \ref{morphisms-lemma-base-change-flat} and \ref{morphisms-lemma-composition-flat}). Thus we can apply Lemma \ref{lemma-recognize-fpqc-covering} in each case to check that our families of morphisms are fpqc coverings. \medskip\noindent Proof of (2). Assume $\{T_i \to T\}_{i\in I}$ is an fpqc covering and for each $i$ we have an fpqc covering $\{f_{ij} : T_{ij} \to T_i\}_{j\in J_i}$. Let $U \subset T$ be an affine open. We can find quasi-compact opens $U_i \subset T_i$ for $i \in I$, almost all empty, such that $U = \bigcup f_i(U_i)$. Then for each $i$ we can choose quasi-compact opens $W_{ij} \subset T_{ij}$ for $j \in J_i$, almost all empty, with $U_i = \bigcup_j f_{ij}(U_{ij})$. Thus $\{T_{ij} \to T\}$ is an fpqc covering. \medskip\noindent Proof of (3). Assume $\{T_i \to T\}_{i\in I}$ is an fpqc covering and $T' \to T$ is a morphism of schemes. Let $U' \subset T'$ be an affine open which maps into the affine open $U \subset T$. Choose quasi-compact opens $U_i \subset T_i$, almost all empty, such that $U = \bigcup f_i(U_i)$. Then $U' \times_U U_i$ is a quasi-compact open of $T' \times_T T_i$ and $U' = \bigcup \text{pr}_1(U' \times_U U_i)$. Since $T'$ can be covered by such affine opens $U' \subset T'$ we see that $\{T' \times_T T_i \to T'\}_{i\in I}$ is an fpqc covering by Lemma \ref{lemma-recognize-fpqc-covering}.
Let $T = \bigcup U_\alpha$ be an affine open covering. For each $\alpha$ the pullback family $\{T_i \times_T U_\alpha \to U_\alpha\}$ can be refined by a standard fpqc covering, hence is an fpqc covering by Lemma \ref{lemma-family-flat-dominated-covering}. As $\{U_\alpha \to T\}$ is an fpqc covering we conclude that $\{T_i \to T\}$ is an fpqc covering by Lemma \ref{lemma-family-flat-fpqc-local-covering}.
Assume (1) and (2) hold. Let $\{f_i : T_i \to T\}_{i \in I}$ be an fpqc covering. Let $s_i \in F(T_i)$ be a family of elements such that $s_i$ and $s_j$ map to the same element of $F(T_i \times_T T_j)$. Let $W \subset T$ be the maximal open subset such that there exists a unique $s \in F(W)$ with $s|_{f_i^{-1}(W)} = s_i|_{f_i^{-1}(W)}$ for all $i$. Such a maximal open exists because $F$ satisfies the sheaf property for Zariski coverings; in fact $W$ is the union of all opens with this property. Let $t \in T$. We will show $t \in W$. To do this we pick an affine open $t \in U \subset T$ and we will show there is a unique $s \in F(U)$ with $s|_{f_i^{-1}(U)} = s_i|_{f_i^{-1}(U)}$ for all $i$. \medskip\noindent By Lemma \ref{lemma-fpqc-affine} we can find a standard fpqc covering $\{U_j \to U\}_{j = 1, \ldots, n}$ refining $\{U \times_T T_i \to U\}$, say by morphisms $h_j : U_j \to T_{i_j}$. By (2) we obtain a unique element $s \in F(U)$ such that $s|_{U_j} = F(h_j)(s_{i_j})$. Note that for any scheme $V \to U$ over $U$ there is a unique section $s_V \in F(V)$ which restricts to $F(h_j \circ \text{pr}_2)(s_{i_j})$ on $V \times_U U_j$ for $j = 1, \ldots, n$. Namely, this is true if $V$ is affine by (2) as $\{V \times_U U_j \to V\}$ is a standard fpqc covering and in general this follows from (1) and the affine case by choosing an affine open covering of $V$. In particular, $s_V = s|_V$. Now, taking $V = U \times_T T_i$ and using that $s_{i_j}|_{T_{i_j} \times_T T_i} = s_i|_{T_{i_j} \times_T T_i}$ we conclude that $s|_{U \times_T T_i} = s_V = s_i|_{U \times_T T_i}$ which is what we had to show. \medskip\noindent Proof of the equivalence of (2) and (2') in the presence of (1). Suppose $\{T_i \to T\}$ is a standard fpqc covering, then $\coprod T_i \to T$ is a faithfully flat morphism of affine schemes. In the presence of (1) we have $F(\coprod T_i) = \prod F(T_i)$ and similarly $F((\coprod T_i) \times_T (\coprod T_i)) = \prod F(T_i \times_T T_{i'})$. Thus the sheaf condition for $\{T_i \to T\}$ and $\{\coprod T_i \to T\}$ is the same.
Let $\{X_i \to X\}_{i = 1, \ldots, n}$ be a standard fpqc covering (Definition \ref{definition-standard-fpqc}). Let $g : \Spec(V) \to X$ be a morphism where $V$ is a valuation ring. Let $x \in X$ be the image of the closed point of $\Spec(V)$. Choose an $i$ and a point $x_i \in X_i$ mapping to $x$. Then $\Spec(V) \times_X X_i$ has a point $x'_i$ mapping to the closed point of $\Spec(V)$. Since $\Spec(V) \times_X X_i \to \Spec(V)$ is flat we can find a specialization $x''_i \leadsto x'_i$ of points of $\Spec(V) \times_X X_i$ with $x''_i$ mapping to the generic point of $\Spec(V)$, see Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat}. By Schemes, Lemma \ref{schemes-lemma-points-specialize} we can choose a valuation ring $W$ and a morphism $h : \Spec(W) \to \Spec(V) \times_X X_i$ such that $h$ maps the generic point of $\Spec(W)$ to $x''_i$ and the closed point of $\Spec(W)$ to $x'_i$. We obtain a commutative diagram $$ \xymatrix{ \Spec(W) \ar[r] \ar[d] & X_i \ar[d] \\ \Spec(V) \ar[r] & X } $$ where $V \to W$ is an extension of valuation rings. This proves the lemma.
Let $T$ be an affine scheme. Let $f : U \to T$ be a proper surjective morphism. Let $U = \bigcup_{j = 1, \ldots, m} U_j$ be a finite affine open covering. We have to show that $\{U_j \to T\}$ is a standard V covering, see Definition \ref{definition-standard-ph-covering}. Let $g : \Spec(V) \to T$ be a morphism where $V$ is a valuation ring with fraction field $K$. Since $U \to T$ is surjective, we may choose a field extension $L/K$ and a commutative diagram $$ \xymatrix{ \Spec(L) \ar[rr] \ar[d] & & U \ar[d] \\ \Spec(K) \ar[r] & \Spec(V) \ar[r]^g & T } $$ By Algebra, Lemma \ref{algebra-lemma-dominate} we can choose a valuation ring $W \subset L$ dominating $V$. By the valuative criterion of properness (Morphisms, Lemma \ref{morphisms-lemma-characterize-proper}) we can then find the morphism $h$ in the commutative diagram $$ \xymatrix{ \Spec(L) \ar[r] \ar[d] & \Spec(W) \ar[r]_h \ar[d] & U \ar[d] \\ \Spec(K) \ar[r] & \Spec(V) \ar[r]^g & X } $$ Since $\Spec(W)$ has a unique closed point, we see that $\Im(h)$ is contained in $U_j$ for some $j$. Thus $h : \Spec(W) \to U_j$ is the desired lift and we conclude $\{U_j \to T\}$ is a standard V covering.
Assertion (1) is clear. \medskip\noindent Proof of (3). Let $U' \subset T'$ be an affine open subscheme. Since $U'$ is quasi-compact we can find a finite affine open covering $U' = U'_1 \cup \ldots \cup U'$ such that $U'_j \to T$ maps into an affine open $U_j \subset T$. Choose a standard V covering $\{U_{jl} \to U_j\}_{l = 1, \ldots, n_j}$ refining $\{T_i \times_T U_j \to U_j\}$. By Lemma \ref{lemma-base-change-standard-V} the base change $\{U_{jl} \times_{U_j} U'_j \to U'_j\}$ is a standard V covering. Note that $\{U'_j \to U'\}$ is a standard V covering (for example by Lemma \ref{lemma-standard-fpqc-standard-V}). By Lemma \ref{lemma-composition-standard-V}  the family $\{U_{jl} \times_{U_j} U'_j \to U'\}$ is a standard V covering. Since $\{U_{jl} \times_{U_j} U'_j \to U'\}$ refines $\{T_i \times_T U' \to U'\}$ we conclude. \medskip\noindent Proof of (2). Let $U \subset T$ be affine open. First we pick a standard V covering $\{U_k \to U\}_{k = 1, \ldots, m}$ refining $\{T_i \times_T U \to U\}$. Say the refinement is given by morphisms $U_k \to T_{i_k}$ over $T$. Then $$ \{T_{i_kj} \times_{T_{i_k}} U_k \to U_k\}_{j \in J_{i_k}} $$ is a V covering by part (3). As $U_k$ is affine, we can find a standard V covering $\{U_{ka} \to U_k\}_{a = 1, \ldots, b_k}$ refining this family. Then we apply Lemma \ref{lemma-composition-standard-V} to see that $\{U_{ka} \to U\}$ is a standard V covering which refines $\{T_{ij} \times_T U \to U\}$. This finishes the proof.
An fpqc covering can affine locally be refined by a standard fpqc covering, see Lemmas \ref{lemma-fpqc-affine}. A standard fpqc covering is a standard V covering, see Lemma \ref{lemma-standard-fpqc-standard-V}. Hence the first statement follows from our definition of V covers in terms of standard V coverings. The conclusion for fppf, syntomic, smooth, \'etale or Zariski coverings follows as these are fpqc coverings, see Lemma \ref{lemma-zariski-etale-smooth-syntomic-fppf-fpqc}. \medskip\noindent The statement on ph coverings follows from Lemma \ref{lemma-standard-ph-standard-V} in the same manner.
Assume (1) and (2) hold. Let $\{f_i : T_i \to T\}_{i \in I}$ be a V covering. Let $s_i \in F(T_i)$ be a family of elements such that $s_i$ and $s_j$ map to the same element of $F(T_i \times_T T_j)$. Let $W \subset T$ be the maximal open subset such that there exists a unique $s \in F(W)$ with $s|_{f_i^{-1}(W)} = s_i|_{f_i^{-1}(W)}$ for all $i$. Such a maximal open exists because $F$ satisfies the sheaf property for Zariski coverings; in fact $W$ is the union of all opens with this property. Let $t \in T$. We will show $t \in W$. To do this we pick an affine open $t \in U \subset T$ and we will show there is a unique $s \in F(U)$ with $s|_{f_i^{-1}(U)} = s_i|_{f_i^{-1}(U)}$ for all $i$. \medskip\noindent We can find a standard V covering $\{U_j \to U\}_{j = 1, \ldots, n}$ refining $\{U \times_T T_i \to U\}$, say by morphisms $h_j : U_j \to T_{i_j}$. By (2) we obtain a unique element $s \in F(U)$ such that $s|_{U_j} = F(h_j)(s_{i_j})$. Note that for any scheme $V \to U$ over $U$ there is a unique section $s_V \in F(V)$ which restricts to $F(h_j \circ \text{pr}_2)(s_{i_j})$ on $V \times_U U_j$ for $j = 1, \ldots, n$. Namely, this is true if $V$ is affine by (2) as $\{V \times_U U_j \to V\}$ is a standard V covering (Lemma \ref{lemma-base-change-standard-V}) and in general this follows from (1) and the affine case by choosing an affine open covering of $V$. In particular, $s_V = s|_V$. Now, taking $V = U \times_T T_i$ and using that $s_{i_j}|_{T_{i_j} \times_T T_i} = s_i|_{T_{i_j} \times_T T_i}$ we conclude that $s|_{U \times_T T_i} = s_V = s_i|_{U \times_T T_i}$ which is what we had to show. \medskip\noindent Proof of the equivalence of (2) and (2') in the presence of (1). Suppose $\{T_i \to T\}_{i = 1, \ldots, n}$ is a standard V covering, then $\coprod_{i = 1, \ldots, n} T_i \to T$ is a morphism of affine schemes which is clearly also a standard V covering. In the presence of (1) we have $F(\coprod T_i) = \prod F(T_i)$ and similarly $F((\coprod T_i) \times_T (\coprod T_i)) = \prod F(T_i \times_T T_{i'})$. Thus the sheaf condition for $\{T_i \to T\}$ and $\{\coprod T_i \to T\}$ is the same.
Assume (1) and let $g : \Spec(V) \to Y$ be as in (2). Since $V$ is a local ring there is an affine open $U \subset Y$ such that $g$ factors through $U$. By Definition \ref{definition-V-covering} we can find a standard V covering $\{U_j \to U\}$ refining $\{X \times_Y U \to U\}$. By Definition \ref{definition-standard-V-covering} we can find a $j$, an extension of valuation rings $V \subset W$ and a commutative diagram $$ \xymatrix{ \Spec(W) \ar[r] \ar[d] & U_j \ar[d] \ar@{..>}[r] & X \ar[ld] \\ \Spec(V) \ar[r] & Y } $$ We have the dotted arrow making the diagram commute by the refinement property of the covering and we see that (2) holds. \medskip\noindent Assume (2) and let $Z \to Y$ and $z' \leadsto z$ be as in (3). By Schemes, Lemma \ref{schemes-lemma-points-specialize} we can find a valuation ring $V$ and a morphism $\Spec(V) \to Z$ such that the closed point of $\Spec(V)$ maps to $z$ and the generic point of $\Spec(V)$ maps to $z'$. By (2) we can find an extension of valuation rings $V \subset W$ and a commutative diagram $$ \xymatrix{ \Spec(W) \ar[rr] \ar[d] & & X \ar[d] \\ \Spec(V) \ar[r] & Z \ar[r] & Y } $$ The generic and closed points of $\Spec(W)$ map to points $w' \leadsto w$ in $Z \times_Y X$ via the induced morphism $\Spec(W) \to Z \times_Y X$. This shows that (3) holds. \medskip\noindent Assume (3) holds and let $U \subset Y$ be an affine open. Choose a finite affine open covering $U \times_Y X = \bigcup_{j = 1, \ldots, m} U_j$. This is possible as $X \to Y$ is quasi-compact. We claim that $\{U_j \to U\}$ is a standard V covering. The claim implies (1) is true and finishes the proof of the lemma. In order to prove the claim, let $V$ be a valuation ring and let $g : \Spec(V) \to U$ be a morphism. By (3) we find a specialization $w' \leadsto w$ of points of $$ T = \Spec(V) \times_X Y = \Spec(V) \times_U (U \times_X Y) $$ such that $w'$ maps to the generic point of $\Spec(V)$ and $w$ maps to the closed point of $\Spec(V)$. By Schemes, Lemma \ref{schemes-lemma-points-specialize} we can find a valuation ring $W$ and a morphism $\Spec(W) \to T$ such that the generic point of $\Spec(W)$ maps to $w'$ and the closed point of $\Spec(W)$ maps to $w$. The composition $\Spec(W) \to T \to \Spec(V)$ corresponds to an inclusion $V \subset W$ which presents $W$ as an extension of the valuation ring $V$. Since $T = \bigcup \Spec(V) \times_U U_j$ is an open covering, we see that $\Spec(W) \to T$ factors through $\Spec(V) \times_U U_j$ for some $j$. Thus we obtain a commutative diagram $$ \xymatrix{ \Spec(W) \ar[d] \ar[r] & U_j \ar[d] \\ \Spec(V) \ar[r] & U } $$ and the proof of the claim is complete.