hcju/stacksqa huggingface dataset:
Consider the presheaf which assigns to $U \in \Ob(S_\etale)$ the empty set if $s|_U \not = t|_U$ and a singleton else. It is clear that this is a subsheaf of the final object of $\Sh(S_\etale)$. By Lemma \ref{lemma-support-subsheaf-final} we find an open $W \subset S$ representing this presheaf. For a geometric point $\overline{x}$ of $S$ we see that $\overline{x} \in W$ if and only if the stalks of $s$ and $t$ at $\overline{x}$ agree. By the description of stalks of pullbacks in Lemma \ref{lemma-stalk-pullback} we see that $W$ has the desired property.
Observe that we have a morphism $i_f : \Sh(T_\tau) \to \Sh(\Sch/S)_\tau)$ such that $\pi_S \circ i_f = f_{small}$ as morphisms $T_\tau \to S_\tau$, see Topologies, Lemmas \ref{topologies-lemma-put-in-T}, \ref{topologies-lemma-morphism-big-small}, \ref{topologies-lemma-put-in-T-etale}, and \ref{topologies-lemma-morphism-big-small-etale}. Since pullback is transitive we see that $i_f^{-1} \pi_S^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$ as desired. \medskip\noindent Let $\{g_i : T_i \to T\}_{i \in I}$ be an fpqc covering. The final statement means the following: Given a sheaf $\mathcal{G}$ on $T_\tau$ and given sections $s_i \in \Gamma(T_i, g_{i, small}^{-1}\mathcal{G})$ whose pullbacks to $T_i \times_T T_j$ agree, there is a unique section $s$ of $\mathcal{G}$ over $T$ whose pullback to $T_i$ agrees with $s_i$. \medskip\noindent Let $V \to T$ be an object of $T_\tau$ and let $t \in \mathcal{G}(V)$. For every $i$ there is a largest open $W_i \subset T_i \times_T V$ such that the pullbacks of $s_i$ and $t$ agree as sections of the pullback of $\mathcal{G}$ to $W_i \subset T_i \times_T V$, see Lemma \ref{lemma-where-sections-are-equal}. Because $s_i$ and $s_j$ agree over $T_i \times_T T_j$ we find that $W_i$ and $W_j$ pullback to the same open over $T_i \times_T T_j \times_T V$. By Descent, Lemma \ref{descent-lemma-open-fpqc-covering} we find an open $W \subset V$ whose inverse image to $T_i \times_T V$ recovers $W_i$.  \medskip\noindent By construction of $g_{i, small}^{-1}\mathcal{G}$ there exists a $\tau$-covering $\{T_{ij} \to T_i\}_{j \in J_i}$, for each $j$ an open immersion or \'etale morphism $V_{ij} \to T$, a section $t_{ij} \in \mathcal{G}(V_{ij})$, and commutative diagrams $$ \xymatrix{ T_{ij} \ar[r] \ar[d] & V_{ij} \ar[d] \\ T_i \ar[r] & T } $$ such that $s_i|_{T_{ij}}$ is the pullback of $t_{ij}$. In other words, after replacing the covering $\{T_i \to T\}$ by $\{T_{ij} \to T\}$ we may assume there are factorizations $T_i \to V_i \to T$ with $V_i \in \Ob(T_\tau)$ and sections $t_i \in \mathcal{G}(V_i)$ pulling back to $s_i$ over $T_i$. By the result of the previous paragraph we find opens $W_i \subset V_i$ such that $t_i|_{W_i}$ ``agrees with'' every $s_j$ over $T_j \times_T W_i$. Note that $T_i \to V_i$ factors through $W_i$. Hence $\{W_i \to T\}$ is a $\tau$-covering and the lemma is proven.
There is a canonical map $\Gamma(S, \mathcal{F}) \to \Gamma(T, f_{small}^{-1}\mathcal{F})$. Since $f$ is surjective (because its fibres are connected) we see that this map is injective. \medskip\noindent To show that the map is surjective, let $\alpha \in \Gamma(T, f_{small}^{-1}\mathcal{F})$. Since $\{T \to S\}$ is an fpqc covering we can use Lemma \ref{lemma-describe-pullback} to see that suffices to prove that $\alpha$ pulls back to the same section over $T \times_S T$ by the two projections. Let $\overline{s} \to S$ be a geometric point. It suffices to show the agreement holds over $(T \times_S T)_{\overline{s}}$ as every geometric point of $T \times_S T$ is contained in one of these geometric fibres. In other words, we are trying to show that $\alpha|_{T_{\overline{s}}}$ pulls back to the same section over $$ (T \times_S T)_{\overline{s}} = T_{\overline{s}} \times_{\overline{s}} T_{\overline{s}} $$ by the two projections to $T_{\overline{s}}$. However, since $\mathcal{F}|_{T_{\overline{s}}}$ is the pullback of $\mathcal{F}|_{\overline{s}}$ it is a constant sheaf with value $\mathcal{F}_{\overline{s}}$. Since $T_{\overline{s}}$ is connected by assumption, any section of a constant sheaf is constant. Hence $\alpha|_{T_{\overline{s}}}$ corresponds to an element of $\mathcal{F}_{\overline{s}}$. Thus the two pullbacks to $(T \times_S T)_{\overline{s}}$ both correspond to this same element and we conclude.
There is a canonical map $\Gamma(S, \mathcal{F}) \to \Gamma(X, f_{small}^{-1}\mathcal{F})$. Since $f$ is surjective (because its fibres are connected) we see that this map is injective. \medskip\noindent To show that the map is surjective, let $\tau \in \Gamma(X, f_{small}^{-1}\mathcal{F})$. It suffices to find an \'etale covering $\{U_i \to S\}$ and sections $\sigma_i \in \mathcal{F}(U_i)$ such that $\sigma_i$ pulls back to $\tau|_{X \times_S U_i}$. Namely, the injectivity shown above guarantees that $\sigma_i$ and $\sigma_j$ restrict to the same section of $\mathcal{F}$ over $U_i \times_S U_j$. Thus we obtain a unique section $\sigma \in \mathcal{F}(S)$ which restricts to $\sigma_i$ over $U_i$. Then the pullback of $\sigma$ to $X$ is $\tau$ because this is true locally. \medskip\noindent Let $\overline{x}$ be a geometric point of $X$ with image $\overline{s}$ in $S$. Consider the image of $\tau$ in the stalk $$ (f_{small}^{-1}\mathcal{F})_{\overline{x}} = \mathcal{F}_{\overline{s}} $$ See Lemma \ref{lemma-stalk-pullback}. We can find an \'etale neighbourhood $U \to S$ of $\overline{s}$ and a section $\sigma \in \mathcal{F}(U)$ mapping to this image in the stalk. Thus after replacing $S$ by $U$ and $X$ by $X \times_S U$ we may assume there exits a section $\sigma$ of $\mathcal{F}$ over $S$ whose image in $(f_{small}^{-1}\mathcal{F})_{\overline{x}}$ is the same as $\tau$. \medskip\noindent By Lemma \ref{lemma-where-sections-are-equal} there exists a maximal open $W \subset X$ such that $f_{small}^{-1}\sigma$ and $\tau$ agree over $W$ and the formation of $W$ commutes with further pullback. Observe that the pullback of $\mathcal{F}$ to the geometric fibre $X_{\overline{s}}$ is the pullback of $\mathcal{F}_{\overline{s}}$ viewed as a sheaf on $\overline{s}$ by $X_{\overline{s}} \to \overline{s}$. Hence we see that $\tau$ and $\sigma$ give sections of the constant sheaf with value $\mathcal{F}_{\overline{s}}$ on $X_{\overline{s}}$ which agree in one point. Since $X_{\overline{s}}$ is connected by assumption, we conclude that $W$ contains $X_s$. The same argument for different geometric fibres shows that $W$ contains every fibre it meets. Since $f$ is submersive, we conclude that $W$ is the inverse image of an open neighbourhood of $s$ in $S$. This finishes the proof.
Follows from Lemma \ref{lemma-sections-upstairs}. Namely, it is clear that $p$ is flat and quasi-compact as the base change of $\Spec(K) \to \Spec(k)$. On the other hand, if $\overline{s} : \Spec(L) \to S$ is a geometric point, then the fibre of $p$ over $\overline{s}$ is the spectrum of $K \otimes_k L$ which is irreducible hence connected by Algebra, Lemma \ref{algebra-lemma-separably-closed-irreducible}.
Note that the assertion makes sense since we have seen that $(X_\etale, \mathcal{O}_{X_\etale})$ and $(Y_\etale, \mathcal{O}_{Y_\etale})$ are locally ringed sites, see Lemma \ref{lemma-etale-site-locally-ringed}. Moreover, we know that $X_\etale$ has enough points, see Theorem \ref{theorem-exactness-stalks} and Remarks \ref{remarks-enough-points}. Hence it suffices to prove that $(f_{small}, f_{small}^\sharp)$ satisfies condition (3) of Modules on Sites, Lemma \ref{sites-modules-lemma-locally-ringed-morphism}. To see this take a point $p$ of $X_\etale$. By Lemma \ref{lemma-points-small-etale-site} $p$ corresponds to a geometric point $\overline{x}$ of $X$. By Lemma \ref{lemma-stalk-pullback} the point $q = f_{small} \circ p$ corresponds to the geometric point $\overline{y} = f \circ \overline{x}$ of $Y$. Hence the assertion we have to prove is that the induced map of stalks $$ \mathcal{O}_{Y, \overline{y}} \longrightarrow \mathcal{O}_{X, \overline{x}} $$ is a local ring map. Suppose that $a \in \mathcal{O}_{Y, \overline{y}}$ is an element of the left hand side which maps to an element of the maximal ideal of the right hand side. Suppose that $a$ is the equivalence class of a triple $(V, \overline{v}, a)$ with $V \to Y$ \'etale, $\overline{v} : \overline{x} \to V$ over $Y$, and $a \in \mathcal{O}(V)$. It maps to the equivalence class of $(X \times_Y V, \overline{x} \times \overline{v}, \text{pr}_V^\sharp(a))$ in the local ring $\mathcal{O}_{X, \overline{x}}$. But it is clear that being in the maximal ideal means that pulling back $\text{pr}_V^\sharp(a)$ to an element of $\kappa(\overline{x})$ gives zero. Hence also pulling back $a$ to $\kappa(\overline{x})$ is zero. Which means that $a$ lies in the maximal ideal of $\mathcal{O}_{Y, \overline{y}}$.
This means that $t : f^{-1}_{small} \to f^{-1}_{small}$ is a transformation of functors such that the diagram $$ \xymatrix{ f_{small}^{-1}\mathcal{O}_Y \ar[rd]_{f_{small}^\sharp}  & & f_{small}^{-1}\mathcal{O}_Y \ar[ll]^t \ar[ld]^{f_{small}^\sharp} \\ & \mathcal{O}_X } $$ is commutative. Suppose $V \to Y$ is \'etale with $V$ affine. By Morphisms, Lemma \ref{morphisms-lemma-quasi-affine-finite-type-over-S} we may choose an immersion $i : V \to \mathbf{A}^n_Y$ over $Y$. In terms of sheaves this means that $i$ induces an injection $h_i : h_V \to \prod_{j = 1, \ldots, n} \mathcal{O}_Y$ of sheaves. The base change $i'$ of $i$ to $X$ is an immersion (Schemes, Lemma \ref{schemes-lemma-base-change-immersion}). Hence $i' : X \times_Y V \to \mathbf{A}^n_X$ is an immersion, which in turn means that $h_{i'} : h_{X \times_Y V} \to \prod_{j = 1, \ldots, n} \mathcal{O}_X$ is an injection of sheaves. Via the identification $f_{small}^{-1}h_V = h_{X \times_Y V}$ of Lemma \ref{lemma-stalk-pullback} the map $h_{i'}$ is equal to $$ \xymatrix{ f_{small}^{-1}h_V \ar[r]^-{f^{-1}h_i} & \prod_{j = 1, \ldots, n} f_{small}^{-1}\mathcal{O}_Y \ar[r]^{\prod f^\sharp} & \prod_{j = 1, \ldots, n} \mathcal{O}_X } $$ (verification omitted). This means that the map $t : f_{small}^{-1}h_V \to f_{small}^{-1}h_V$ fits into the commutative diagram $$ \xymatrix{ f_{small}^{-1}h_V \ar[r]^-{f^{-1}h_i} \ar[d]^t & \prod_{j = 1, \ldots, n} f_{small}^{-1}\mathcal{O}_Y \ar[r]^-{\prod f^\sharp} \ar[d]^{\prod t} & \prod_{j = 1, \ldots, n} \mathcal{O}_X \ar[d]^{\text{id}}\\ f_{small}^{-1}h_V \ar[r]^-{f^{-1}h_i} & \prod_{j = 1, \ldots, n} f_{small}^{-1}\mathcal{O}_Y \ar[r]^-{\prod f^\sharp} & \prod_{j = 1, \ldots, n} \mathcal{O}_X } $$ The commutativity of the right square holds by our assumption on $t$ explained above. Since the composition of the horizontal arrows is injective by the discussion above we conclude that the left vertical arrow is the identity map as well. Any sheaf of sets on $Y_\etale$ admits a surjection from a (huge) coproduct of sheaves of the form $h_V$ with $V$ affine (combine Lemma \ref{lemma-alternative} with Sites, Lemma \ref{sites-lemma-sheaf-coequalizer-representable}). Thus we conclude that $t : f_{small}^{-1} \to f_{small}^{-1}$ is the identity transformation as desired.
Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Let $U$ be an object of $X_\etale$. By assumption we can find a covering $\{W_i \to U\}$ in $X_\etale$ such that each $W_i$ is an open and closed subscheme of $X \times_Y V_i$ for some object $V_i$ of $Y_\etale$. The sheaf condition shows that $$ \mathcal{F}(U) \subset \prod \mathcal{F}(W_i) $$ and that $\mathcal{F}(W_i)$ is a direct summand of $\mathcal{F}(X \times_Y V_i) = f_{small, *}\mathcal{F}(V_i)$. Hence it is clear that $f_{small, *}$ reflects injections. \medskip\noindent Next, suppose that $a : \mathcal{G} \to \mathcal{F}$ is a map of abelian sheaves such that $f_{small, *}a$ is surjective. Let $s \in \mathcal{F}(U)$ with $U$ as above. With $W_i$, $V_i$ as above we see that it suffices to show that $s|_{W_i}$ is \'etale locally the image of a section of $\mathcal{G}$ under $a$. Since $\mathcal{F}(W_i)$ is a direct summand of $\mathcal{F}(X \times_Y V_i)$ it suffices to show that for any $V \in \Ob(Y_\etale)$ any element $s \in \mathcal{F}(X \times_Y V)$ is \'etale locally on $X \times_Y V$ the image of a section of $\mathcal{G}$ under $a$. Since $\mathcal{F}(X \times_Y V) = f_{small, *}\mathcal{F}(V)$ we see by assumption that there exists a covering $\{V_j \to V\}$ such that $s$ is the image of $s_j \in f_{small, *}\mathcal{G}(V_j) = \mathcal{G}(X \times_Y V_j)$. This proves $f_{small, *}$ reflects surjections. \medskip\noindent Parts (2), (3) follow formally from part (1), see Modules on Sites, Lemma \ref{sites-modules-lemma-reflect-surjections}.
Let $U \to X$ be an \'etale morphism and $u \in U$. The geometric statement (A) reduces directly to the case where $U$ and $Y$ are affine schemes. Denote $x \in X$ and $y \in Y$ the images of $u$. Since $X \to Y$ is locally quasi-finite, and $U \to X$ is locally quasi-finite (see Morphisms, Lemma \ref{morphisms-lemma-etale-locally-quasi-finite}) we see that $U \to Y$ is locally quasi-finite (see Morphisms, Lemma \ref{morphisms-lemma-composition-quasi-finite}). Moreover both $X \to Y$ and $U \to Y$ are separated. Thus More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant} applies to both morphisms. This means we may pick an \'etale neighbourhood $(V, v) \to (Y, y)$ such that $$ X \times_Y V = W \amalg R, \quad U \times_Y V = W' \amalg R' $$ and points $w \in W$, $w' \in W'$ such that \begin{enumerate} \item $W$, $R$ are open and closed in $X \times_Y V$, \item $W'$, $R'$ are open and closed in $U \times_Y V$, \item $W \to V$ and $W' \to V$ are finite, \item $w$, $w'$ map to $v$, \item $\kappa(v) \subset \kappa(w)$ and $\kappa(v) \subset \kappa(w')$ are purely inseparable, and \item no other point of $W$ or $W'$ maps to $v$. \end{enumerate} Here is a commutative diagram $$ \xymatrix{ U \ar[d] & U \times_Y V \ar[l] \ar[d] & W' \amalg R' \ar[d] \ar[l] \\ X \ar[d] & X \times_Y V \ar[l] \ar[d] & W \amalg R \ar[l] \\ Y & V \ar[l] } $$ After shrinking $V$ we may assume that $W'$ maps into $W$: just remove the image the inverse image of $R$ in $W'$; this is a closed set (as $W' \to V$ is finite) not containing $v$. Then $W' \to W$ is finite because both $W \to V$ and $W' \to V$ are finite. Hence $W' \to W$ is finite \'etale, and there is exactly one point in the fibre over $w$ with $\kappa(w) = \kappa(w')$. Hence $W' \to W$ is an isomorphism in an open neighbourhood $W^\circ$ of $w$, see \'Etale Morphisms, Lemma \ref{etale-lemma-finite-etale-one-point}. Since $W \to V$ is finite the image of $W \setminus W^\circ$ is a closed subset $T$ of $V$ not containing $v$. Thus after replacing $V$ by $V \setminus T$ we may assume that $W' \to W$ is an isomorphism. Now the decomposition $X \times_Y V = W \amalg R$ and the morphism $W \to U$ are as desired and we win.
Let $U \to X$ be \'etale, and let $u \in U$ be a point. We have to find $V \to Y$ \'etale, a disjoint union decomposition $X \times_Y V = W \amalg W'$ and an $X$-morphism $W \to U$ with $u$ in the image. We may shrink $U$ and $Y$ and assume $U$ and $Y$ are affine. In this case also $X$ is affine, since an integral morphism is affine by definition. Write $Y = \Spec(A)$, $X = \Spec(B)$ and $U = \Spec(C)$. Then $A \to B$ is an integral ring map, and $B \to C$ is an \'etale ring map. By Algebra, Lemma \ref{algebra-lemma-etale} we can find a finite $A$-subalgebra $B' \subset B$ and an \'etale ring map $B' \to C'$ such that $C = B \otimes_{B'} C'$. Thus the question reduces to the \'etale morphism $U' = \Spec(C') \to X' = \Spec(B')$ over the finite morphism $X' \to Y$. In this case the result follows from Lemma \ref{lemma-locally-quasi-finite-A}.
Consider $V \to Y$ \'etale, $\{U_i \to X \times_Y V\}$ an \'etale covering, and $v \in V$. We have to find a $V' \to V$ and decomposition and maps as in Lemma \ref{lemma-simplify-B}. We may shrink $V$ and $Y$, hence we may assume that $V$ and $Y$ are affine. Since $X$ is finite over $Y$, this also implies that $X$ is affine. During the proof we may (finitely often) replace $(V, v)$ by an \'etale neighbourhood $(V', v')$ and correspondingly the covering $\{U_i \to X \times_Y V\}$ by $\{V' \times_V U_i \to X \times_Y V'\}$. \medskip\noindent Since $X \times_Y V \to V$ is finite there exist finitely many (pairwise distinct) points $x_1, \ldots, x_n \in X \times_Y V$ mapping to $v$. We may apply More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant} to $X \times_Y V \to V$ and the points $x_1, \ldots, x_n$ lying over $v$ and find an \'etale neighbourhood $(V', v') \to (V, v)$ such that $$ X \times_Y V' = R \amalg \coprod T_a $$ with $T_a \to V'$ finite with exactly one point $p_a$ lying over $v'$ and moreover $\kappa(v') \subset \kappa(p_a)$ purely inseparable, and such that $R \to V'$ has empty fibre over $v'$. Because $X \to Y$ is finite, also $R \to V'$ is finite. Hence after shrinking $V'$ we may assume that $R = \emptyset$. Thus we may assume that $X \times_Y V = X_1 \amalg \ldots \amalg X_n$ with exactly one point $x_l \in X_l$ lying over $v$ with moreover $\kappa(v) \subset \kappa(x_l)$ purely inseparable. Note that this property is preserved under refinement of the \'etale neighbourhood $(V, v)$. \medskip\noindent For each $l$ choose an $i_l$ and a point $u_l \in U_{i_l}$ mapping to $x_l$. Now we apply property (A) for the finite morphism $X \times_Y V \to V$ and the \'etale morphisms $U_{i_l} \to X \times_Y V$ and the points $u_l$. This is permissible by Lemma \ref{lemma-integral-A} This gives produces an \'etale neighbourhood $(V', v') \to (V, v)$ and decompositions $$ X \times_Y V' = W_l \amalg R_l $$ and $X$-morphisms $a_l : W_l \to U_{i_l}$ whose image contains $u_{i_l}$. Here is a picture: $$ \xymatrix{ & & & U_{i_l} \ar[d] & \\ W_l \ar[rrru] \ar[r] & W_l \amalg R_l \ar@{=}[r] & X \times_Y V' \ar[r] \ar[d] & X \times_Y V \ar[r] \ar[d] & X \ar[d] \\ & & V' \ar[r] & V \ar[r] & Y } $$ After replacing $(V, v)$ by $(V', v')$ we conclude that each $x_l$ is contained in an open and closed neighbourhood $W_l$ such that the inclusion morphism $W_l \to X \times_Y V$ factors through $U_i \to X \times_Y V$ for some $i$. Replacing $W_l$ by $W_l \cap X_l$ we see that these open and closed sets are disjoint and moreover that $\{x_1, \ldots, x_n\} \subset W_1 \cup \ldots \cup W_n$. Since $X \times_Y V \to V$ is finite we may shrink $V$ and assume that $X \times_Y V = W_1 \amalg \ldots \amalg W_n$ as desired.
Consider $V \to Y$ \'etale, $\{U_i \to X \times_Y V\}$ an \'etale covering, and $v \in V$. We have to find a $V' \to V$ and decomposition and maps as in Lemma \ref{lemma-simplify-B}. We may shrink $V$ and $Y$, hence we may assume that $V$ and $Y$ are affine. Since $X$ is integral over $Y$, this also implies that $X$ and $X \times_Y V$ are affine. We may refine the covering $\{U_i \to X \times_Y V\}$, and hence we may assume that $\{U_i \to X \times_Y V\}_{i = 1, \ldots, n}$ is a standard \'etale covering. Write $Y = \Spec(A)$, $X = \Spec(B)$, $V = \Spec(C)$, and $U_i = \Spec(B_i)$. Then $A \to B$ is an integral ring map, and $B \otimes_A C \to B_i$ are \'etale ring maps. By Algebra, Lemma \ref{algebra-lemma-etale} we can find a finite $A$-subalgebra $B' \subset B$ and an \'etale ring map $B' \otimes_A C \to B'_i$ for $i = 1, \ldots, n$ such that $B_i = B \otimes_{B'} B'_i$. Thus the question reduces to the \'etale covering $\{\Spec(B'_i) \to X' \times_Y V\}_{i = 1, \ldots, n}$ with $X' = \Spec(B')$ finite over $Y$. In this case the result follows from Lemma \ref{lemma-finite-B}.
Let $g : U \to X$ be an \'etale morphism. We need to find an object $V \to Y$ of $Y_\etale$ and a surjective morphism $X \times_Y V \to U$ over $X$. Suppose that for every $u \in U$ we can find an object $V_u \to Y$ of $Y_\etale$ and a morphism $h_u : X \times_Y V_u \to U$ over $X$ with $u \in \Im(h_u)$. Then we can take $V = \coprod V_u$ and $h = \coprod h_u$ and we win. Hence given a point $u \in U$ we find a pair $(V_u, h_u)$ as above. To do this we may shrink $U$ and assume that $U$ is affine. In this case $g : U \to X$ is locally quasi-finite. Let $g^{-1}(g(\{u\})) = \{u, u_2, \ldots, u_n\}$. Since there are no specializations $u_i \leadsto u$ we may replace $U$ by an affine neighbourhood so that $g^{-1}(g(\{u\})) = \{u\}$. \medskip\noindent The image $g(U) \subset X$ is open, hence $f(g(U))$ is locally closed in $Y$. Choose an open $V \subset Y$ such that $f(g(U)) = f(X) \cap V$. It follows that $g$ factors through $X \times_Y V$ and that the resulting $\{U \to X \times_Y V\}$ is an \'etale covering. Since $f$ has property (B) , see Lemma \ref{lemma-integral-B}, we see that there exists an \'etale covering $\{V_j \to V\}$ such that $X \times_Y V_j \to X \times_Y V$ factor through $U$. This implies that $V' = \coprod V_j$ is \'etale over $Y$ and that there is a morphism $h : X \times_Y V' \to U$ whose image surjects onto $g(U)$. Since $u$ is the only point in its fibre it must be in the image of $h$ and we win.
By Lemmas \ref{lemma-integral-A}, \ref{lemma-integral-B} and \ref{lemma-integral-homeo-onto-image-C} we know that the morphism $f$ has properties (A), (B), and (C). Moreover, by Lemma \ref{lemma-property-C-closed-implies} we know that the functor $Y_\etale \to X_\etale$ is almost cocontinuous. Now we have \begin{enumerate} \item property (C) implies (1) by Lemma \ref{lemma-property-C-implies}, \item almost continuous implies (2) by Sites, Lemma \ref{sites-lemma-morphism-of-sites-almost-cocontinuous}, \item property (B) implies (3) by Lemma \ref{lemma-property-B-implies}. \end{enumerate} Properties (4), (5), and (6) follow formally from the first three, see Sites, Lemma \ref{sites-lemma-exactness-properties} and Modules on Sites, Lemma \ref{sites-modules-lemma-exactness}. Property (7) we saw above.
Consider the scheme $M = U \times_Y U'$. The graph $\Gamma_h \subset M_Z$ of $h$ is open. This is true for example as $\Gamma_h$ is the image of a section of the \'etale morphism $\text{pr}_{1, Z} : M_Z \to U_Z$, see \'Etale Morphisms, Proposition \ref{etale-proposition-properties-sections}. Hence there exists an open subscheme $W \subset M$ whose intersection with the closed subset $M_Z$ is $\Gamma_h$. Set $a = \text{pr}_1|_W$ and $b = \text{pr}_2|_W$.
Note that $i_{small, *}\mathcal{G}|_{U_\etale} = *$ is the final object in the category of \'etale sheaves on $U$, i.e., the sheaf which associates a singleton set to each scheme \'etale over $U$. This explains the value of $(i_{small, *}\mathcal{G})_{\overline{x}}$ if $\overline{x} \not \in Z$. \medskip\noindent Next, suppose that $\overline{x} \in Z$. Note that $$ (i_{small, *}\mathcal{G})_{\overline{x}} = \colim_{(U, \overline{u})} \mathcal{G}(U_Z) $$ and on the other hand $$ \mathcal{G}_{\overline{x}} = \colim_{(V, \overline{v})} \mathcal{G}(V). $$ Let $\mathcal{C}_1 = \{(U, \overline{u})\}^{opp}$ be the opposite of the category of \'etale neighbourhoods of $\overline{x}$ in $X$, and let $\mathcal{C}_2 = \{(V, \overline{v})\}^{opp}$ be the opposite of the category of \'etale neighbourhoods of $\overline{x}$ in $Z$. The canonical map $$ \mathcal{G}_{\overline{x}} \longrightarrow (i_{small, *}\mathcal{G})_{\overline{x}} $$ corresponds to the functor $F : \mathcal{C}_1 \to \mathcal{C}_2$, $F(U, \overline{u}) = (U_Z, \overline{x})$. Now Lemmas \ref{lemma-closed-immersion-almost-essentially-surjective} and \ref{lemma-closed-immersion-almost-full} imply that $\mathcal{C}_1$ is cofinal in $\mathcal{C}_2$, see Categories, Definition \ref{categories-definition-cofinal}. Hence it follows that the displayed arrow is an isomorphism, see Categories, Lemma \ref{categories-lemma-cofinal}.
Let us prove the lemma when $\tau = syntomic$. The question is local on $U$. Thus we may assume that $U$ is an affine scheme mapping into an affine of $Y$. Hence we reduce to proving the following case: $Y = \Spec(A)$, $X = \Spec(A/I)$, and $U = \Spec(\overline{B})$, where $A/I \to \overline{B}$ be a syntomic ring map. By Algebra, Lemma \ref{algebra-lemma-lift-syntomic} we can find elements $\overline{g}_i \in \overline{B}$ such that $\overline{B}_{\overline{g}_i} = A_i/IA_i$ for certain syntomic ring maps $A \to A_i$. This proves the lemma in the syntomic case. The proof of the smooth case is the same except it uses Algebra, Lemma \ref{algebra-lemma-lift-smooth}. In the \'etale case use Algebra, Lemma \ref{algebra-lemma-lift-etale}.