hcju/stacksqa huggingface dataset:
Let $U \to X$ be an \'etale surjective morphism, where $U$ is a scheme. Let $R = U \times_X U$. The open subspaces of $X$ correspond $1 - 1$ with open subschemes of $U$ which are $R$-invariant. Hence there is a set of them. Let $X_i$, $i \in I$ be the set of open subspaces of $X$ which are schemes, i.e., are representable. Consider the open subspace $X' \subset X$ whose underlying set of points is the open $\bigcup |X_i|$ of $|X|$. By Lemma \ref{lemma-characterize-surjective} we see that $$ \coprod X_i \longrightarrow X' $$ is a surjective map of sheaves on $(\Sch/S)_{fppf}$. But since each $X_i \to X'$ is representable by open immersions we see that in fact the map is surjective in the Zariski topology. Namely, if $T \to X'$ is a morphism from a scheme into $X'$, then $X_i \times_X' T$ is an open subscheme of $T$. Hence we can apply Schemes, Lemma \ref{schemes-lemma-glue-functors} to see that $X'$ is a scheme.
Let $X' \subset X$ be the maximal open subspace which is a scheme (Lemma \ref{lemma-subscheme}). Let $x \in |X|$ be a point of codimension $0$ on $X$. By Lemma \ref{lemma-codimension-0-points-dense} it suffices to show $x \in X'$. Let $U \to X$ be as in the statement of the lemma. Write $R = U \times_X U$ and denote $s, t : R \to U$ the projections as usual. Note that $s, t$ are surjective, finite and \'etale. By Lemma \ref{lemma-finite-fibres-presentation} the fibre of $|U| \to |X|$ over $x$ is finite, say $\{\eta_1, \ldots, \eta_n\}$. By Lemma \ref{lemma-codimension-0-points} each $\eta_i$ is the generic point of an irreducible component of $U$. By Properties, Lemma \ref{properties-lemma-maximal-points-affine} we can find an affine open $W \subset U$ containing $\{\eta_1, \ldots, \eta_n\}$ (this is where we use that $U$ is quasi-separated). By Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine} we may assume that $W$ is $R$-invariant. Since $W \subset U$ is an $R$-invariant affine open, the restriction $R_W$ of $R$ to $W$ equals $R_W = s^{-1}(W) = t^{-1}(W)$ (see Groupoids, Definition \ref{groupoids-definition-invariant-open} and discussion following it). In particular the maps $R_W \to W$ are finite \'etale also. It follows that $R_W$ is affine. Thus we see that $W/R_W$ is a scheme, by Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence}. On the other hand, $W/R_W$ is an open subspace of $X$ by Spaces, Lemma \ref{spaces-lemma-finding-opens} and it contains $x$ by construction.
We first show that $X/G$ is a scheme. Since the action is free the morphism $j = (a, \text{pr}) : G \times_S X \to X \times_S X$ is a monomorphism and hence an equivalence relation, see Groupoids, Lemma \ref{groupoids-lemma-free-action}. The maps $s, t : G \times_S X \to X$ are finite locally free as we've assumed that $G \to S$ is finite locally free. To conclude it now suffices to prove the last assumption of Proposition \ref{proposition-finite-flat-equivalence-global} holds. Since the action of $G$ is over $S$ it suffices to prove that any finite set of points in a fibre of $X \to S$ is contained in an affine open of $X$. If $X$ is isomorphic to an open subscheme of an affine scheme or isomorphic to an open subscheme of $\text{Proj}(A)$ for some graded ring $A$ this follows from Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine}. If $X \to S$ is affine, or quasi-affine, or projective, or quasi-projective, we may replace $S$ by an affine open and we get back to the case we just dealt with. If $G \to S$ is radicial, then the orbits of points on $X$ under the action of $G$ are singletons and the condition trivially holds. Some details omitted. \medskip\noindent To see that $X \to X/G$ is an fppf $G$-torsor (Groupoids, Definition \ref{groupoids-definition-principal-homogeneous-space}) we have to show that $G \times_S X \to X \times_{X/G} X$ is an isomorphism and that $X \to X/G$ fppf locally has sections. The second part is clear from the fact that $X \to X/G$ is surjective as a map of fppf sheaves (by construction). The first part follows from the isomorphism $R = U \times_M U$ in the conclusion of Proposition \ref{proposition-finite-flat-equivalence-global} (note that $R = G \times_S X$ in our case).
Since $M$ represents the quotient sheaf we have a cartesian diagram $$ \xymatrix{ R \ar[r]_-j \ar[d] & U \times_S U \ar[d] \\ M \ar[r] & M \times_S M } $$ of schemes. Since $U \times_S U \to M \times_S M$ is surjective finite locally free, to show that $M \to M \times_S M$ is quasi-compact, resp.\ a closed immersion, it suffices to show that $j : R \to U \times_S U$ is quasi-compact, resp.\ a closed immersion, see Descent, Lemmas \ref{descent-lemma-descending-property-quasi-compact} and \ref{descent-lemma-descending-property-closed-immersion}. Since $j : R \to U \times_S U$ is a morphism over $U$ and since $R$ is finite over $U$, we see that $j$ is quasi-compact as soon as the projection $U \times_S U \to U$ is quasi-separated (Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence}). Since $j$ is a monomorphism and locally of finite type, we see that $j$ is a closed immersion as soon as it is proper (\'Etale Morphisms, Lemma \ref{etale-lemma-characterize-closed-immersion}) which will be the case as soon as the projection $U \times_S U \to U$ is separated (Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). This proves (1) and (3). To prove (2) and (4) we replace $S$ by $\Spec(\mathbf{Z})$, see Definition \ref{definition-separated}. Since Lemma \ref{lemma-quotient-scheme} is proved through an application of Proposition \ref{proposition-finite-flat-equivalence-global} the final statement is clear too.
Combining Topology, Lemma \ref{topology-lemma-sober-local} and Lemma \ref{lemma-quasi-separated-quasi-compact-pieces} we see that we may assume that there exists an affine scheme $U$ and a surjective, quasi-compact, \'etale morphism $U \to X$. Set $R = U \times_X U$ with projection maps $s, t : R \to U$. Applying Lemma \ref{lemma-finite-fibres-presentation} we see that the fibres of $s, t$ are finite. It follows all the assumptions of Topology, Lemma \ref{topology-lemma-quotient-kolmogorov} are met, and we conclude that $|X|$ is Kolmogorov\footnote{ Actually we use here also Schemes, Lemma \ref{schemes-lemma-scheme-sober} (soberness schemes), Morphisms, Lemmas \ref{morphisms-lemma-etale-flat} and \ref{morphisms-lemma-generalizations-lift-flat} (generalizations lift along \'etale morphisms), Lemma \ref{lemma-points-presentation} (points on an algebraic space in terms of a presentation), and Lemma \ref{lemma-topology-points} (openness quotient map).}. \medskip\noindent It remains to show that every irreducible closed subset $T \subset |X|$ has a generic point. By Lemma \ref{lemma-reduced-closed-subspace} there exists a closed subspace $Z \subset X$ with $|Z| = |T|$. Note that $U \times_X Z \to Z$ is a quasi-compact, surjective, \'etale morphism from an affine scheme to $Z$, hence $Z$ is Zariski locally quasi-separated by Lemma \ref{lemma-quasi-separated-quasi-compact-pieces}. By Proposition \ref{proposition-locally-quasi-separated-open-dense-scheme} we see that there exists an open dense subspace $Z' \subset Z$ which is a scheme. This means that $|Z'| \subset T$ is open dense. Hence the topological space $|Z'|$ is irreducible, which means that $Z'$ is an irreducible scheme. By Schemes, Lemma \ref{schemes-lemma-scheme-sober} we conclude that $|Z'|$ is the closure of a single point $\eta \in |Z'| \subset T$ and hence also $T = \overline{\{\eta\}}$, and we win.
By Topology, Definition \ref{topology-definition-spectral-space} we have to check that $|X|$ is sober, quasi-compact, has a basis of quasi-compact opens, and the intersection of any two quasi-compact opens is quasi-compact. By Lemma \ref{lemma-quasi-separated-sober} we see that $|X|$ is sober. By Lemma \ref{lemma-quasi-compact-space} we see that $|X|$ is quasi-compact. By Lemma \ref{lemma-quasi-compact-affine-cover} there exists an affine scheme $U$ and a surjective \'etale morphism $f : U \to X$. Since $|f| : |U| \to |X|$ is open and continuous and since $|U|$ has a basis of quasi-compact opens, we conclude that $|X|$ has a basis of quasi-compact opens. Finally, suppose that $A, B \subset |X|$ are quasi-compact open. Then $A = |X'|$ and $B = |X''|$ for some open subspaces $X', X'' \subset X$ (Lemma \ref{lemma-open-subspaces}) and we can choose affine schemes $V$ and $W$ and surjective \'etale morphisms $V \to X'$ and $W \to X''$ (Lemma \ref{lemma-quasi-compact-affine-cover}). Then $A \cap B$ is the image of $|V \times_X W| \to |X|$ (Lemma \ref{lemma-points-cartesian}). Since $V \times_X W$ is quasi-compact as $X$ is quasi-separated (Lemma \ref{lemma-characterize-quasi-separated}) we conclude that $A \cap B$ is quasi-compact and the proof is finished.
Set $R = \Spec(k) \times_X \Spec(k)$, so that we have a fibre product diagram $$ \xymatrix{ R \ar[r]_-s \ar[d]_-t & \Spec(k) \ar[d] \\ \Spec(k) \ar[r] & X } $$ By Spaces, Lemma \ref{spaces-lemma-space-presentation} we know $X = \Spec(k)/R$ is the quotient sheaf. Because $\Spec(k) \to X$ is \'etale, the morphisms $s$ and $t$ are \'etale. Hence $R = \coprod_{i \in I} \Spec(k_i)$ is a disjoint union of spectra of fields, and both $s$ and $t$ induce finite separable field extensions $s, t : k \subset k_i$, see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}. Because $$ R = \Spec(k) \times_X \Spec(k) = (\Spec(k) \times_S \Spec(k)) \times_{X \times_S X, \Delta} X $$ and since $\Delta$ is quasi-compact by assumption we conclude that $R \to \Spec(k) \times_S \Spec(k)$ is quasi-compact. Hence $R$ is quasi-compact as $\Spec(k) \times_S \Spec(k)$ is affine. We conclude that $I$ is finite. This implies that $s$ and $t$ are finite locally free morphisms. Hence by Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence} we conclude that $\Spec(k)/R$ is represented by $\Spec(k')$, with $k' \subset k$ finite locally free where $$ k' = \{x \in k \mid s_i(x) = t_i(x)\text{ for all }i \in I\} $$ It is easy to see that $k'$ is a field.
Recall that our definition of an \'etale morphism from a scheme into an algebraic space comes from Spaces, Definition \ref{spaces-definition-relative-representable-property} via the fact that any morphism from a scheme into an algebraic space is representable. \medskip\noindent Part (1) of the lemma follows from this, the fact that \'etale morphisms are preserved under composition (Morphisms, Lemma \ref{morphisms-lemma-composition-etale}) and Spaces, Lemmas \ref{spaces-lemma-composition-representable-transformations-property} and \ref{spaces-lemma-morphism-schemes-gives-representable-transformation-property} (which are formal). \medskip\noindent To prove part (2) choose a scheme $W$ over $S$ and a surjective \'etale morphism $W \to X$. Consider the base change $\chi_W : W \times_X U \to W \times_X U'$ of $\chi$. As $W \times_X U$ and $W \times_X U'$ are \'etale over $W$, we conclude that $\chi_W$ is \'etale, by Morphisms, Lemma \ref{morphisms-lemma-etale-permanence-two}. On the other hand, in the commutative diagram $$ \xymatrix{ W \times_X U \ar[r] \ar[d] & W \times_X U' \ar[d] \\ U \ar[r] & U' } $$ the two vertical arrows are \'etale and surjective. Hence by Descent, Lemma \ref{descent-lemma-syntomic-smooth-etale-permanence} we conclude that $U \to U'$ is \'etale. \medskip\noindent To prove part (2) choose a scheme $W$ over $S$ and a morphism $W \to X$. As above we consider the diagram $$ \xymatrix{ W \times_X U \ar[r] \ar[d] & W \times_X U' \ar[d] \ar[r] & W \ar[d] \\ U \ar[r] & U' \ar[r] & X } $$ Now we know that $W \times_X U \to W \times_X U'$ is surjective \'etale (as a base change of $U \to U'$) and that $W \times_X U \to W$ is \'etale. Thus $W \times_X U' \to W$ is \'etale by Descent, Lemma \ref{descent-lemma-syntomic-smooth-etale-permanence}. By definition this means that $\varphi'$ is \'etale.
Let us prove that (4) implies (1). Assume a diagram as in (4) given. Let $W \to X$ be an \'etale morphism with $W$ a scheme. Then we see that $W \times_X U \to U$ is \'etale. Hence $W \times_X U \to V$ is \'etale as the composition of the \'etale morphisms of schemes $W \times_X U \to U$ and $U \to V$. Therefore $W \times_X U \to Y$ is \'etale by Lemma \ref{lemma-etale-over-space} (1). Since also the projection $W \times_X U \to W$ is surjective and \'etale, we conclude from Lemma \ref{lemma-etale-over-space} (3) that $W \to Y$ is \'etale. \medskip\noindent Let us prove that (1) implies (4). Assume (1). Choose a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U \to X$ and $V \to Y$ are surjective and \'etale, see Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}. By assumption the morphism $U \to Y$ is \'etale, and hence $U \to V$ is \'etale by Lemma \ref{lemma-etale-over-space} (2). \medskip\noindent We omit the proof that (2) and (3) are also equivalent to (1).
Let $X \to Y$ be an \'etale morphism of algebraic spaces over $S$. Let $Z \to Y$ be a morphism of algebraic spaces. Choose a scheme $U$ and a surjective \'etale morphism $U \to X$. Choose a scheme $W$ and a surjective \'etale morphism $W \to Z$. Then $U \to Y$ is \'etale, hence in the diagram $$ \xymatrix{ W \times_Y U \ar[d] \ar[r] & W \ar[d] \\ Z \times_Y X \ar[r] & Z } $$ the top horizontal arrow is \'etale. Moreover, the left vertical arrow is surjective and \'etale (verification omitted). Hence we conclude that the lower horizontal arrow is \'etale by Lemma \ref{lemma-etale-local}.
Choose a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r]_\chi & V \ar[d] \\ X \ar[r] & Y } $$ where $U \to X$ and $V \to Y$ are surjective and \'etale, see Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}. By assumption the morphisms $\varphi : U \to X \to Z$ and $\psi : V \to Y \to Z$ are \'etale. Moreover, $\psi \circ \chi = \varphi$ by our assumption on $f, g, h$. Hence $U \to V$ is \'etale by Lemma \ref{lemma-etale-over-space} part (2).
Let $U$ be an affine scheme, and let $U \to X$ be an \'etale morphism. By Definition \ref{definition-etale} we see that $U \to \Spec(k)$ is an \'etale morphism. Hence $U = \coprod_{i = 1, \ldots, n} \Spec(k_i)$ is a finite disjoint union of spectra of finite separable extensions $k_i$ of $k$, see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}. The $R = U \times_X U \to U \times_{\Spec(k)} U$ is a monomorphism and $U \times_{\Spec(k)} U$ is also a finite disjoint union of spectra of finite separable extensions of $k$. Hence by Schemes, Lemma \ref{schemes-lemma-mono-towards-spec-field} we see that $R$ is similarly a finite disjoint union of spectra of finite separable extensions of $k$. This $U$ and $R$ are affine and both projections $R \to U$ are finite locally free. Hence $U/R$ is a scheme by Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence}. By Spaces, Lemma \ref{spaces-lemma-finding-opens} it is also an open subspace of $X$. By Lemma \ref{lemma-subscheme} we conclude that $X$ is a scheme.
We have to show that the functor satisfies the assumptions (1) -- (5) of Sites, Lemma \ref{sites-lemma-equivalence}. It is clear that the functor is continuous and cocontinuous, which proves assumptions (1) and (2). Assumptions (3) and (4) hold simply because the functor is fully faithful. Assumption (5) holds, because an algebraic space by definition has a covering by a scheme.
Let us show that the functor described in (1) satisfies the assumptions of Sites, Proposition \ref{sites-proposition-get-morphism}. Thus we have to show that $Y_{spaces, \etale}$ has a final object (namely $Y$) and that the functor transforms this into a final object in $X_{spaces, \etale}$ (namely $X$). This is clear as $X \times_Y Y = X$ in any category. Next, we have to show that $Y_{spaces, \etale}$ has fibre products. This is true since the category of algebraic spaces has fibre products, and since $V \times_Y V'$ is \'etale over $Y$ if $V$ and $V'$ are \'etale over $Y$ (see Lemmas \ref{lemma-composition-etale} and \ref{lemma-base-change-etale} above). OK, so the proposition applies and we see that we get a morphism of sites as described in (1). \medskip\noindent Part (2) you get by unwinding the definitions. Part (3) is clear by using the equivalences for $X$ and $Y$ from Lemma \ref{lemma-compare-etale-sites} above. Part (4) follows, because if $f$ is representable, then the functors above fit into a commutative diagram $$ \xymatrix{ X_\etale \ar[r] & X_{spaces, \etale} \\ Y_\etale \ar[r] \ar[u] & Y_{spaces, \etale} \ar[u] } $$ of categories.
Note that (1) and (2) are the same because the functors $f_{small, *}$ and $f_{small}^{-1}$ are a pair of adjoint functors. Suppose that $\alpha : f_{small}^{-1}\mathcal{G} \to \mathcal{F}$ is a map of sheaves on $Y_\etale$. Let a diagram $$ \xymatrix{ U \ar[d]_g \ar[r]_{j_U} & X \ar[d]^f \\ V \ar[r]^{j_V} & Y } $$ as in Definition \ref{definition-f-map} be given. By the commutativity of the diagram we also get a map $g_{small}^{-1}(j_V)^{-1}\mathcal{G} \to (j_U)^{-1}\mathcal{F}$ (compare Sites, Section \ref{sites-section-localize} for the description of the localization functors). Hence we certainly get a map $\varphi_{(V, U, g)} : \mathcal{G}(V) = (j_V)^{-1}\mathcal{G}(V) \to (j_U)^{-1}\mathcal{F}(U) = \mathcal{F}(U)$. We omit the verification that this rule is compatible with further restrictions and defines an $f$-map from $\mathcal{G}$ to $\mathcal{F}$. \medskip\noindent Conversely, suppose that we are given an $f$-map $\varphi = (\varphi_{(U, V, g)})$. Let $\mathcal{G}'$ (resp.\ $\mathcal{F}'$) denote the extension of $\mathcal{G}$ (resp.\ $\mathcal{F}$) to $Y_{spaces, \etale}$ (resp.\ $X_{spaces, \etale}$), see Lemma \ref{lemma-compare-etale-sites}. Then we have to construct a map of sheaves $$ \mathcal{G}' \longrightarrow (f_{spaces, \etale})_*\mathcal{F}' $$ To do this, let $V \to Y$ be an \'etale morphism of algebraic spaces. We have to construct a map of sets $$ \mathcal{G}'(V) \to \mathcal{F}'(X \times_Y V) $$ Choose an \'etale surjective morphism $V' \to V$ with $V'$ a scheme, and after that choose an \'etale surjective morphism $U' \to X \times_U V'$ with $U'$ a scheme. We get a morphism of schemes $g' : U' \to V'$ and also a morphism of schemes $$ g'' : U' \times_{X \times_Y V} U' \longrightarrow V' \times_V V' $$ Consider the following diagram $$ \xymatrix{ \mathcal{F}'(X \times_Y V) \ar[r] & \mathcal{F}(U') \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(U' \times_{X \times_Y V} U') \\ \mathcal{G}'(X \times_Y V) \ar[r] \ar@{..>}[u] & \mathcal{G}(V') \ar@<1ex>[r] \ar@<-1ex>[r] \ar[u]_{\varphi_{(U', V', g')}} & \mathcal{G}(V' \times_V V') \ar[u]_{\varphi_{(U'', V'', g'')}} } $$ The compatibility of the maps $\varphi_{...}$ with restriction shows that the two right squares commute. The definition of coverings in $X_{spaces, \etale}$ shows that the horizontal rows are equalizer diagrams. Hence we get the dotted arrow. We leave it to the reader to show that these arrows are compatible with the restriction mappings.
Note that by our very definition of an \'etale morphism of algebraic spaces (Definition \ref{definition-etale}) it is indeed the case that the rule given defines a functor $j$ as indicated. It is clear that $j$ is cocontinuous and continuous, simply because a covering $\{U_i \to U\}$ of $j(\varphi : U \to X)$ in $Y_\etale$ is the same thing as a covering of $(\varphi : U \to X)$ in $X_\etale$. It remains to show that $j$ induces the same morphism of topoi as $f_{small}$. To see this we consider the diagram $$ \xymatrix{ X_\etale \ar[r] \ar[d]^j & X_{spaces, \etale} \ar@/_/[d]_{j_{spaces}} \\ Y_\etale \ar[r] & Y_{spaces, \etale} \ar@/_/[u]_{v : V \mapsto X \times_Y V} } $$ of categories. Here the functor $j_{spaces}$ is the obvious extension of $j$ to the category $X_{spaces, \etale}$. Thus the inner square is commutative. In fact $j_{spaces}$ can be identified with the localization functor $j_X : Y_{spaces, \etale}/X \to Y_{spaces, \etale}$ discussed in Sites, Section \ref{sites-section-localize}. Hence, by Sites, Lemma \ref{sites-lemma-localize-given-products} the cocontinuous functor $j_{spaces}$ and the functor $v$ of the diagram induce the same morphism of topoi. By Sites, Lemma \ref{sites-lemma-composition-cocontinuous} the commutativity of the inner square (consisting of cocontinuous functors between sites) gives a commutative diagram of associated morphisms of topoi. Hence, by the construction of $f_{small}$ in Lemma \ref{lemma-functoriality-etale-site} we win.
Consider the following diagram of functors $$ \xymatrix{ X'_{spaces, \etale} \ar[r]_j & X_{spaces, \etale} \\ Y'_{spaces, \etale} \ar[r]^j \ar[u]^{V' \mapsto V' \times_{Y'} X'} & Y_{spaces, \etale} \ar[u]_{V \mapsto V \times_Y X} } $$ The horizontal arrows are localizations and the vertical arrows induce morphisms of sites. Hence the last statement of Sites, Lemma \ref{sites-lemma-localize-morphism} gives (1). To see (2) apply (1) to an injective resolution of $\mathcal{F}$ and use that restriction is exact and preserves injectives (see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}).
Given a sheaf $\mathcal{F}$ on $X_\etale$ and an object $\varphi : U \to X$ of $X_\etale$ we set $\mathcal{F}_U = \varphi_{small}^{-1}\mathcal{F}$. If $\varphi' : U' \to X$ is a second object of $X_\etale$, and $f : U' \to U$ is a morphism between them, then the isomorphism $c_f$ comes from the fact that $f_{small}^{-1} \circ \varphi_{small}^{-1} = (\varphi')^{-1}_{small}$, see Lemma \ref{lemma-functoriality-etale-site}. The condition on the transitivity of the isomorphisms $c_f$ follows from the functoriality of the small \'etale sites also; verification omitted. \medskip\noindent Conversely, suppose we are given a collection of data $(\mathcal{F}_U, c_f)$ as in the lemma. In this case we simply define $\mathcal{F}$ by the rule $U \mapsto \mathcal{F}_U(U)$. Details omitted.
[First proof of Lemma \ref{lemma-descent-sheaf}] Let $\mathcal{C} = X_{spaces, \etale}$. By Lemma \ref{lemma-etale-morphism-topoi} and its proof we have $U_{spaces, \etale} = \mathcal{C}/U$ and the pullback functor $\varphi^{-1}$ is just the restriction functor. Moreover, $\{U \to X\}$ is a covering of the site $\mathcal{C}$ and $R = U \times_X U$. The isomorphism $\alpha$ is just the canonical identification $$ \left(\mathcal{F}|_{\mathcal{C}/U}\right)|_{\mathcal{C}/U \times_X U} = \left(\mathcal{F}|_{\mathcal{C}/U}\right)|_{\mathcal{C}/U \times_X U} $$ and the commutativity of the diagram is the cocycle condition for glueing data. Hence this lemma is a special case of glueing of sheaves, see Sites, Section \ref{sites-section-glueing-sheaves}.
For part (1), consider the fibre product $U = U_1 \times_X U_2$. It is \'etale over both $U_1$ and $U_2$ because \'etale morphisms are preserved under base change and composition, see Lemmas \ref{lemma-base-change-etale} and \ref{lemma-composition-etale}. The map $\overline{u} \to U$ defined by $(\overline{u}_1, \overline{u}_2)$ gives it the structure of an \'etale neighborhood mapping to both $U_1$ and $U_2$. \medskip\noindent For part (2), define $U''$ as the fibre product $$ \xymatrix{ U'' \ar[r] \ar[d] & U \ar[d]^{(h_1, h_2)} \\ U' \ar[r]^-\Delta & U' \times_X U'. } $$ Since $\overline{u}$ and $\overline{u}'$ agree over $X$ with $\overline{x}$, we see that $\overline{u}'' = (\overline{u}, \overline{u}')$ is a geometric point of $U''$. In particular $U'' \not = \emptyset$. Moreover, since $U'$ is \'etale over $X$, so is the fibre product $U'\times_X U'$ (as seen above in the case of $U_1 \times_X U_2$). Hence the vertical arrow $(h_1, h_2)$ is \'etale by Lemma \ref{lemma-etale-permanence}. Therefore $U''$ is \'etale over $U'$ by base change, and hence also \'etale over $X$ (because compositions of \'etale morphisms are \'etale). Thus $(U'', \overline{u}'')$ is a solution to the problem posed by (2). \medskip\noindent To see the final assertion, choose any surjective \'etale morphism $U' \to U$ where $U'$ is a scheme. Then $U' \times_U \overline{u}$ is a scheme surjective and \'etale over $\overline{u} = \Spec(k)$ with $k$ algebraically closed. It follows (see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}) that $U' \times_U \overline{u} \to \overline{u}$ has a section which gives us the desired $\overline{u}'$.