hcju/stacksqa huggingface dataset:
Choose an affine scheme $U'$ with $u' \in U'$ and an \'etale morphism $U' \to U$ which maps $u'$ to $u$. If we can prove the lemma for $(U', u') \to (X, x)$ then the lemma follows. Hence we may assume that $U$ is a scheme, in particular that $U \to X$ is representable. Then look at the cartesian diagram $$ \xymatrix{ \Spec(k) \times_{\overline{x}, X, \varphi} U \ar[d]_{\text{pr}_1} \ar[r]_-{\text{pr}_2} & U \ar[d]^\varphi \\ \Spec(k) \ar[r]^-{\overline{x}} & X } $$ The projection $\text{pr}_1$ is the base change of an \'etale morphisms so it is \'etale, see Lemma \ref{lemma-base-change-etale}. Therefore, the scheme $\Spec(k) \times_{\overline{x}, X, \varphi} U$ is a disjoint union of finite separable extensions of $k$, see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}. But $k$ is algebraically closed, so all these extensions are trivial, so $\Spec(k) \times_{\overline{x}, X, \varphi} U$ is a disjoint union of copies of $\Spec(k)$ and each of these corresponds to a geometric point $\overline{u}$ with $\varphi \circ \overline{u} = \overline{x}$. By Lemma \ref{lemma-points-cartesian} the map $$ |\Spec(k) \times_{\overline{x}, X, \varphi} U| \longrightarrow |\Spec(k)| \times_{|X|} |U| $$ is surjective, hence we can pick $\overline{u}$ to lie over $u$.
Let $u \in |U|$ be the image of $\overline{u}$. As $|U| = \bigcup_{i \in I} \varphi_i(|U_i|)$ there exists an $i$ and a point $u_i \in U_i$ mapping to $x$. Apply Lemma \ref{lemma-geometric-lift-to-usual} to $(U_i, u_i) \to (U, u)$ and $\overline{u}$ to get the desired geometric point.
In the proof of Lemma \ref{lemma-geometric-lift-to-cover} we have seen that the scheme $U_{\overline{x}}$ is a disjoint union of schemes isomorphic to $\overline{x}$. Thus we can also think of $|U_{\overline{x}}|$ as the set of geometric points of $U$ lying over $\overline{x}$, i.e., as the collection of morphisms $\overline{u} : \overline{x} \to U$ fitting into the diagram of Definition \ref{definition-geometric-point}. From this it follows that $u(X)$ is a singleton, and that $u(U \times_V W) = u(U) \times_{u(V)} u(W)$ whenever $U \to V$ and $W \to V$ are morphisms in $X_\etale$. And, given a covering $\{U_i \to U\}_{i \in I}$ in $X_\etale$ we see that $\coprod u(U_i) \to u(U)$ is surjective by Lemma \ref{lemma-geometric-lift-to-cover}. Hence Sites, Proposition \ref{sites-proposition-point-limits} applies, so $p$ is a point of the site $X_\etale$. Finally, the our functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is given by exactly the same colimit as the functor $\mathcal{F} \mapsto \mathcal{F}_p$ associated to $p$ in Sites, Equation \ref{sites-equation-stalk} which proves the final assertion.
This result follows from the general material in Modules on Sites, Section \ref{sites-modules-section-stalks}. This is true because $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ comes from a point of the small \'etale site of $X$, see Lemma \ref{lemma-stalk-gives-point}. See the proof of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-stalk-exact} for a direct proof of some of these statements in the setting of the small \'etale site of a scheme.
Recall that $f_{small}$ is defined via $f_{spaces, small}$ in Lemma \ref{lemma-functoriality-etale-site}. Parts (1), (2) and (3) are general consequences of the fact that $f_{spaces, \etale} : X_{spaces, \etale} \to Y_{spaces, \etale}$ is a morphism of sites, see Sites, Definition \ref{sites-definition-morphism-sites} for (2), Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact} for (1), and Sites, Lemma \ref{sites-lemma-pullback-representable-sheaf} for (3). \medskip\noindent Proof of (4). This statement is a special case of Sites, Lemma \ref{sites-lemma-point-morphism-sites} via Lemma \ref{lemma-stalk-gives-point}. We also provide a direct proof. Note that by Lemma \ref{lemma-stalk-exact}. taking stalks commutes with sheafification. Let $\mathcal{G}'$ be the sheaf on $Y_{spaces, \etale}$ whose restriction to $Y_\etale$ is $\mathcal{G}$. Recall that $f_{spaces, \etale}^{-1}\mathcal{G}'$ is the sheaf associated to the presheaf $$ U \longrightarrow \colim_{U \to X \times_Y V} \mathcal{G}'(V), $$ see Sites, Sections \ref{sites-section-continuous-functors} and \ref{sites-section-functoriality-PSh}. Thus we have \begin{align*} (f_{spaces, \etale}^{-1}\mathcal{G}')_{\overline{x}} & = \colim_{(U, \overline{u})} f_{spaces, \etale}^{-1}\mathcal{G}'(U) \\ & = \colim_{(U, \overline{u})} \colim_{a : U \to X \times_Y V} \mathcal{G}'(V) \\ & = \colim_{(V, \overline{v})} \mathcal{G}'(V) \\ & = \mathcal{G}'_{\overline{y}} \end{align*} in the third equality the pair $(U, \overline{u})$ and the map $a : U \to X \times_Y V$ corresponds to the pair $(V, a \circ \overline{u})$. Since the stalk of $\mathcal{G}'$ (resp.\ $f_{spaces, \etale}^{-1}\mathcal{G}'$) agrees with the stalk of $\mathcal{G}$ (resp.\ $f_{small}^{-1}\mathcal{G}$), see Equation (\ref{equation-stalk-spaces-etale}) the result follows.
By Sites, Lemma \ref{sites-lemma-point-site-topos} there is a one to one correspondence between points of the site and points of the associated topos. Hence we may assume that $p$ is given by a functor $u : X_\etale \to \textit{Sets}$ which defines a point of the site $X_\etale$. Let $U \in \Ob(X_\etale)$ be an object whose structure morphism $j : U \to X$ is surjective. Note that $h_U$ is a sheaf which surjects onto the final sheaf. Since taking stalks is exact we see that $(h_U)_p = u(U)$ is not empty (use Sites, Lemma \ref{sites-lemma-points-recover}). Pick $x \in u(U)$. By Sites, Lemma \ref{sites-lemma-point-localize} we obtain a point $q : \Sh(pt) \to \Sh(U_\etale)$ such that $p = j_{small} \circ q$, so that $\mathcal{F}_p = (\mathcal{F}|_U)_q$ functorially. By \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-points-small-etale-site} there is a geometric point $\overline{u}$ of $U$ and a functorial isomorphism $\mathcal{G}_q = \mathcal{G}_{\overline{u}}$ for $\mathcal{G} \in \Sh(U_\etale)$. Set $\overline{x} = j \circ \overline{u}$. Then we see that $\mathcal{F}_{\overline{x}} \cong (\mathcal{F}|_U)_{\overline{u}}$ functorially in $\mathcal{F}$ on $X_\etale$ by Lemma \ref{lemma-stalk-pullback} and we win.
The condition means that $\mathcal{F}(U)$ is a singleton or empty for all $\varphi : U \to X$ in $\Ob(X_{spaces, \etale})$. In particular local sections always glue. If $\mathcal{F}(U) \not = \emptyset$, then $\mathcal{F}(\varphi(U)) \not = \emptyset$ because $\varphi(U) \subset X$ is an open subspace (Lemma \ref{lemma-etale-open}) and $\{\varphi : U \to \varphi(U)\}$ is a covering in $X_{spaces, \etale}$. Take $W = \bigcup_{\varphi : U \to S, \mathcal{F}(U) \not = \emptyset} \varphi(U)$ to conclude.
Since $\mathcal{F}$ is a sheaf in the \'etale topology the restriction of $\mathcal{F}$ to $U_{Zar}$ is a sheaf on $U$ in the Zariski topology. Hence there exists a Zariski open $W$ having property (1), see Modules, Lemma \ref{modules-lemma-support-section-closed}. Let $\varphi : V \to U$ be an arrow of $X_\etale$. Note that $\varphi(V) \subset U$ is an open subspace (Lemma \ref{lemma-etale-open}) and that $\{V \to \varphi(V)\}$ is an \'etale covering. Hence if $\sigma|_V = 0$, then by the sheaf condition for $\mathcal{F}$ we see that $\sigma|_{\varphi(V)} = 0$. This proves (2). To prove (3) we have to show that if $(U, \overline{u}, \sigma)$ defines the zero element of $\mathcal{F}_{\overline{s}}$, then $\overline{u} \in W$. This is true because the assumption means there exists a morphism of \'etale neighbourhoods $(V, \overline{v}) \to (U, \overline{u})$ such that $\sigma|_V = 0$. Hence by (2) we see that $V \to U$ maps into $W$, and hence $\overline{u} \in W$.
Part (1) holds by definition. Parts (2) and (3) hold because they holds for the restriction of $\mathcal{F}$ and $\mathcal{G}$ to $U_{Zar}$, see Modules, Lemma \ref{modules-lemma-support-section-closed}. Part (4) is a direct consequence of Lemma \ref{lemma-zero-over-image} part (3). Parts (5) and (6) follow from the other parts.
By Lemma \ref{lemma-f-map} it suffices to give an $f$-map from $\mathcal{O}_Y$ to $\mathcal{O}_X$. In other words, for every commutative diagram $$ \xymatrix{ U \ar[d]_g \ar[r] & X \ar[d]^f \\ V \ar[r] & Y } $$ where $U \in X_\etale$, $V \in Y_\etale$ we have to give a map of rings $ (f^\sharp)_{(U, V, g)} : \Gamma(V, \mathcal{O}_V) \to \Gamma(U, \mathcal{O}_U). $ Of course we just take $(f^\sharp)_{(U, V, g)} = g^\sharp$. It is clear that this is compatible with restriction mappings and hence indeed gives an $f$-map. We omit checking compatibility with compositions and agreement with the construction in Descent, Remark \ref{descent-remark-change-topologies-ringed}.
The equivalence of (1) and (2) follows from Properties, Lemma \ref{properties-lemma-characterize-reduced} applied to affine schemes \'etale over $X$. The final statements follow the cited lemma and fact that $\Gamma(X, \mathcal{O}_X)$ is a subring of $\Gamma(U, \mathcal{O}_U)$ for some reduced scheme $U$ \'etale over $X$.
We know that the structure sheaf $\mathcal{O}_U$ on $U_\etale$ is the restriction of the structure sheaf of $X$. Hence the first equality follows from Lemma \ref{lemma-stalk-pullback} part (4). The second equality is explained in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-etale-local-ring}.
This follows because the stalks $\mathcal{O}_{X, \overline{x}}$ are local, and because $S_\etale$ has enough points, see Lemmas \ref{lemma-describe-etale-local-ring} and Theorem \ref{theorem-exactness-stalks}. See Modules on Sites, Lemma \ref{sites-modules-lemma-locally-ringed-stalk} and \ref{sites-modules-lemma-ringed-stalk-not-zero} for the fact that this implies the small \'etale site is locally ringed.
The equivalence of (2) and (3) follows from the fact that the isomorphism type of $\mathcal{O}_{X, \overline{x}}$ only depends on $x \in |X|$, see Remark \ref{remark-map-stalks}. Using Lemma \ref{lemma-describe-etale-local-ring} the equivalence of (1) and (2)$+$(3) comes down to the following statement: Given any local ring $R$ we have $\dim(R) = \dim(R^{sh})$. This is More on Algebra, Lemma \ref{more-algebra-lemma-henselization-dimension}.
Choose a scheme $U$ and a point $u \in U$ and an \'etale morphism $U \to X$ which maps $u$ to $x$. Then the composition $U \to Y$ is also \'etale and maps $u$ to $f(x)$. Thus the statements (1) and (2) follow as the relevant integers are defined in terms of the behaviour of the scheme $U$ at $u$. See Definition \ref{definition-dimension-at-point} for (1). Part (3) is an immediate consequence of (1), see Definition \ref{definition-dimension}.
The equivalence of (2) and (3) follows from the fact that the isomorphism type of $\mathcal{O}_{X, \overline{x}}$ only depends on $x \in |X|$, see Remark \ref{remark-map-stalks}. Using Lemma \ref{lemma-describe-etale-local-ring} the equivalence of (1) and (2)$+$(3) comes down to the following statement: a local ring is reduced if and only if its strict henselization is reduced. This is More on Algebra, Lemma \ref{more-algebra-lemma-henselization-reduced}.
The equivalence of (1) and (2) follows from the fact that irreducible components of $U$ passing through $u$ are in $1$-$1$ correspondence with minimal primes of the local ring of $U$ at $u$. Let $a : U \to X$ and $u \in U$ be as in (1). Then $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of $\mathcal{O}_{U, u}$ by Lemma \ref{lemma-describe-etale-local-ring}. In particular (4) and (5) are equivalent by More on Algebra, Lemma \ref{more-algebra-lemma-geometrically-unibranch}. The equivalence of (2), (3), and (4) follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-nr-branches}.
The equivalence of (1) and (2) follows from the fact that irreducible components of $U$ passing through $u$ are in $1$-$1$ correspondence with minimal primes of the local ring of $U$ at $u$. Let $a : U \to X$ and $u \in U$ be as in (1). Then $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of $\mathcal{O}_{U, u}$ by Lemma \ref{lemma-describe-etale-local-ring}. Recall that the (geometric) number of branches of $U$ at $u$ is the number of minimal prime ideals of the (strict) henselization of $\mathcal{O}_{U, u}$. In particular (4) and (5) are equivalent. The equivalence of (2), (3), and (4) follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-nr-branches}.
Assume $X$ is locally Noetherian. Choose a scheme $U$ and a surjective \'etale morphism $U \to X$. As $X$ is locally Noetherian we see that $U$ is locally Noetherian. By Properties, Lemma \ref{properties-lemma-Noetherian-topology} this means that $|U|$ is a locally Noetherian topological space. Since $|U| \to |X|$ is open and surjective we conclude that $|X|$ is locally Noetherian by Topology, Lemma \ref{topology-lemma-image-Noetherian}. This proves (1). If $X$ is quasi-compact and locally Noetherian, then $|X|$ is quasi-compact and locally Noetherian. Hence $|X|$ is Noetherian by Topology, Lemma \ref{topology-lemma-quasi-compact-locally-Noetherian-Noetherian}.
Choose an \'etale neighbourhood $(U, \overline{u})$ of $\overline{x}$ where $U$ is a scheme. Then $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of the local ring of $U$ at $u$, see Lemma \ref{lemma-describe-etale-local-ring}. By our definition of Noetherian spaces the scheme $U$ is Noetherian. Hence we conclude by More on Algebra, Lemma \ref{more-algebra-lemma-henselization-noetherian}.
Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. By assumption $U$ is locally Noetherian. Moreover, every \'etale local ring $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of a local ring on $U$ and conversely, see Lemma \ref{lemma-describe-etale-local-ring}. Thus by More on Algebra, Lemma \ref{more-algebra-lemma-henselization-regular} we see that (2) is equivalent to every local ring of $U$ being regular, i.e., $U$ being a regular scheme (see Properties, Lemma \ref{properties-lemma-characterize-regular}). This equivalent to (1) by Definition \ref{definition-type-property}.
Let $U$ be a scheme, $u \in U$ a point, and let $a : U \to X$ be an \'etale morphism mapping $u$ to $x$. For any geometric point $\overline{x}$ of $X$ lying over $x$, the \'etale local ring $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of a local ring on $U$ at $u$, see Lemma \ref{lemma-describe-etale-local-ring}. Thus we conclude by More on Algebra, Lemma \ref{more-algebra-lemma-henselization-regular}.
By the description of inverse image in Lemma \ref{lemma-etale-morphism-topoi} and the definition of the structure sheaves it is clear that $f_{small}^{-1}\mathcal{O}_Y = \mathcal{O}_X$. Since the pullback $$ f^*\mathcal{G} = f_{small}^{-1}\mathcal{G} \otimes_{f_{small}^{-1}\mathcal{O}_Y} \mathcal{O}_X $$ by definition we conclude that $f^*\mathcal{G} = f_{small}^{-1}\mathcal{G}$. The exactness is clear because $f_{small}^{-1}$ is exact, as $f_{small}$ is a morphism of topoi.
The morphism $(f_{spaces, \etale, c}, f_c^\sharp)$ is defined as a composition $f' \circ j$ of a localization and a base change map. Similarly $g$ is a composition $U \to V \times_Y X \to V$. Hence it suffices to prove the lemma in the following two cases: (1) $f = \text{id}$, and (2) $U = X \times_Y V$. In case (1) the morphism $g : U \to V$ is \'etale, see Lemma \ref{lemma-etale-permanence}. Hence $(g_{spaces, \etale}, g^\sharp)$ is a localization morphism by the discussion surrounding Equations (\ref{equation-localize}) and (\ref{equation-localize-ringed}) which is exactly the content of the lemma in this case. In case (2) the morphism $g_{spaces, \etale}$ comes from the morphism of ringed sites given by the functor $V_{spaces, \etale} \to U_{spaces, \etale}$, $V'/V \mapsto V' \times_V U/U$ which is also what the morphism $f'$ is defined by, see Sites, Lemma \ref{sites-lemma-localize-morphism}. We omit the verification that $(f')^\sharp = g^\sharp$ in this case (both are the restriction of $f^\sharp$ to $U_{spaces, \etale}$).
Note that $(g_{small}, g^\sharp)$ is $2$-isomorphic as a morphism of ringed topoi to the morphism of ringed topoi associated to the morphism of ringed sites $(g_{spaces, \etale}, g^\sharp)$. Hence we conclude by Lemma \ref{lemma-relocalize-morphism} and Modules on Sites, Lemma \ref{sites-modules-lemma-relocalize-morphism-compare}.
Let us prove that $X$ is a sheaf for the fppf topology. Namely, suppose that $\{g_i : T_i \to T\}$ is a covering of $(\Sch/S)_{fppf}$ and $(y_i, s_i) \in X(T_i)$ satisfy the glueing condition, i.e., the restriction of $(y_i, s_i)$ and $(y_j, s_j)$ to $T_i \times_T T_j$ agree. Then since $Y$ is a sheaf for the fppf topology, we see that the $y_i$ give rise to a unique morphism $y : T \to Y$ such that $y_i = y \circ g_i$. Then we see that $y_{i, small}^{-1}\mathcal{F} = g_{i, small}^{-1}y_{small}^{-1}\mathcal{F}$. Hence the sections $s_i$ glue uniquely to a section of $y_{small}^{-1}\mathcal{F}$ by \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback}. \medskip\noindent The construction that sends $\mathcal{F} \in \Ob(\Sh(Y_\etale))$ to $X \in \Ob((\Sch/S)_{fppf})$ preserves finite limits and all colimits since each of the functors $y_{small}^{-1}$ have this property. Of course, if $V \in \Ob(Y_\etale)$, then the construction sends the representable sheaf $h_V$ on $Y_\etale$ to the representable functor represented by $V$. \medskip\noindent By Sites, Lemma \ref{sites-lemma-sheaf-coequalizer-representable} we can find a set $I$, for each $i \in I$ an object $V_i$ of $Y_\etale$ and a surjective map of sheaves $$ \coprod h_{V_i} \longrightarrow \mathcal{F} $$ on $Y_\etale$. The set theoretic condition we need is that the index set $I$ is not too large\footnote{It suffices if the supremum of the cardinalities of the stalks of $\mathcal{F}$ at geometric points of $Y$ is bounded by the size of some object of $(\Sch/S)_{fppf}$.}. Then $V = \coprod V_i$ is an object of $(\Sch/S)_{fppf}$ and therefore an object of $Y_\etale$ and we have a surjective map $h_V \to \mathcal{F}$. \medskip\noindent Observe that the product of $h_V$ with itself in $\Sh(Y_\etale)$ is $h_{V \times_Y V}$. Consider the fibre product $$ h_V \times_\mathcal{F} h_V \subset h_{V \times_Y V} $$ There is an open subscheme $R$ of $V \times_Y V$ such that $h_V \times_\mathcal{F} h_V = h_R$, see Lemma \ref{lemma-support-subsheaf-final} (small detail omitted). By the Yoneda lemma we obtain two morphisms $s, t : R \to V$ in $Y_\etale$ and we find a coequalizer diagram $$ \xymatrix{ h_R \ar@<1ex>[r] \ar@<-1ex>[r] & h_V \ar[r] & \mathcal{F} } $$ in $\Sh(Y_\etale)$. Of course the morphisms $s, t$ are \'etale and define an \'etale equivalence relation $(t, s) : R \to V \times_S V$. \medskip\noindent By the discussion in the preceding two paragraphs we find a coequalizer diagram $$ \xymatrix{ R \ar@<1ex>[r] \ar@<-1ex>[r] & V \ar[r] & X } $$ in $(\Sch/S)_{fppf}$. Thus $X = V/R$ is an algebraic space by Spaces, Theorem \ref{spaces-theorem-presentation}. The other statements follow readily from this; details omitted.
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}. Hence it suffices to prove that $(f_{small}, f^\sharp)$ 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 \'etale local rings $$ \mathcal{O}_{Y, \overline{y}} \longrightarrow \mathcal{O}_{X, \overline{x}} $$ is a local ring map. You can prove this directly, but instead we deduce it from the corresponding result for schemes. To do this choose a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r] & Y } $$ where $U$ and $V$ are schemes, and the vertical arrows are surjective \'etale (see Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}). Choose a lift $\overline{u} : \overline{x} \to U$ (possible by Lemma \ref{lemma-geometric-lift-to-cover}). Set $\overline{v} = \psi \circ \overline{u}$. We obtain a commutative diagram of \'etale local rings $$ \xymatrix{ \mathcal{O}_{U, \overline{u}} & \mathcal{O}_{V, \overline{v}} \ar[l] \\ \mathcal{O}_{X, \overline{x}} \ar[u] & \mathcal{O}_{Y, \overline{y}}. \ar[l] \ar[u] } $$ By \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-morphism-locally-ringed} the top horizontal arrow is a local ring map. Finally by Lemma \ref{lemma-describe-etale-local-ring} the vertical arrows are isomorphisms. Hence we win.
Let $X'$, resp.\ $Y'$ be $X$ viewed as an algebraic space over $\Spec(\mathbf{Z})$, see Spaces, Definition \ref{spaces-definition-base-change}. It is clear from the construction that $(X_{small}, \mathcal{O})$ is equal to $(X'_{small}, \mathcal{O})$ and similarly for $Y$. Hence we may work with $X'$ and $Y'$. In other words we may assume that $S = \Spec(\mathbf{Z})$. \medskip\noindent Assume $S = \Spec(\mathbf{Z})$, $f : X \to Y$ and $t$ are as in the lemma. This means that $t : f^{-1}_{small} \to f^{-1}_{small}$ is a transformation of functors such that the diagram $$ \xymatrix{ f_{small}^{-1}\mathcal{O}_Y \ar[rd]_{f^\sharp}  & & f_{small}^{-1}\mathcal{O}_Y \ar[ll]^t \ar[ld]^{f^\sharp} \\ & \mathcal{O}_X } $$ is commutative. Suppose $V \to Y$ is \'etale with $V$ affine. Write $V = \Spec(B)$. Choose generators $b_j \in B$, $j \in J$ for $B$ as a $\mathbf{Z}$-algebra. Set $T = \Spec(\mathbf{Z}[\{x_j\}_{j \in J}])$. In the following we will use that $\Mor_{\Sch}(U, T) = \prod_{j \in J} \Gamma(U, \mathcal{O}_U)$ for any scheme $U$ without further mention. The surjective ring map $\mathbf{Z}[x_j] \to B$, $x_j \mapsto b_j$ corresponds to a closed immersion $V \to T$. We obtain a monomorphism $$ i : V \longrightarrow T_Y = T \times Y $$ of algebraic spaces over $Y$. In terms of sheaves on $Y_\etale$ the morphism $i$ induces an injection $h_i : h_V \to \prod_{j \in J} \mathcal{O}_Y$ of sheaves. The base change $i' : X \times_Y V \to T_X$ of $i$ to $X$ is a monomorphism too (Spaces, Lemma \ref{spaces-lemma-base-change-representable-transformations-property}). Hence $i' : X \times_Y V \to T_X$ is a monomorphism, which in turn means that $h_{i'} : h_{X \times_Y V} \to \prod_{j \in J} \mathcal{O}_X$ 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 $$ \xymatrix{ f_{small}^{-1}h_V \ar[r]^-{f^{-1}h_i} & \prod_{j \in J} f_{small}^{-1}\mathcal{O}_Y \ar[r]^{\prod f^\sharp} & \prod_{j \in J} \mathcal{O}_X } $$ (verification omitted). This means that the map $t : f_{small}^{-1}h_V \to f_{small}^{-1}h_V$ fits into the commutative diagram $$ \xymatrix{ f_{small}^{-1}h_V \ar[r]^-{f^{-1}h_i} \ar[d]^t & \prod_{j \in J} f_{small}^{-1}\mathcal{O}_Y \ar[r]^-{\prod f^\sharp} \ar[d]^{\prod t} & \prod_{j \in J} \mathcal{O}_X \ar[d]^{\text{id}}\\ f_{small}^{-1}h_V \ar[r]^-{f^{-1}h_i} & \prod_{j \in J} f_{small}^{-1}\mathcal{O}_Y \ar[r]^-{\prod f^\sharp} & \prod_{j \in J} \mathcal{O}_X } $$ 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.