hcju/stacksqa huggingface dataset:
Let $X_0 \subset X = \Spec(A)$ and $Y_0 \subset Y = \Spec(B)$ be the sets of closed points. By assumption $Y_0$ maps into $X_0$ and the induced map $Y_0 \to X_0$ is a bijection. As a space $\Spec(A)$ is the disjoint union of the spectra of the local rings of $A$ at closed points. Similarly for $B$. Hence $X \to Y$ is a bijection. Since $A \to B$ is flat we have going down (Algebra, Lemma \ref{algebra-lemma-flat-going-down}). Thus Algebra, Lemma \ref{algebra-lemma-unique-prime-over-localize-below} shows for any prime $\mathfrak q \subset B$ lying over $\mathfrak p \subset A$ we have $B_\mathfrak q = B_\mathfrak p$. Since $B_\mathfrak q = A_\mathfrak p$ by assumption, we see that $A_\mathfrak p = B_\mathfrak p$ for all primes $\mathfrak p$ of $A$. Thus $A = B$ by Algebra, Lemma \ref{algebra-lemma-characterize-zero-local}.
Set $X = \Spec(A)$ and $Y = \Spec(B)$. Let $X_0 \subset X$ and $Y_0 \subset Y$ be the set of closed points. Let $A \to A'$ be the ind-Zariski morphism of affine schemes such that with $X' = \Spec(A')$ the diagram $$ \xymatrix{ X' \ar[r] \ar[d] & \pi_0(X') \ar[d] \\ X \ar[r] & \pi_0(X) } $$ is cartesian in the category of topological spaces and such that $\pi_0(X') = \pi_0(Y)$ as spaces over $\pi_0(X)$, see Lemma \ref{lemma-construct-profinite}. By Lemma \ref{lemma-silly} we see that $X'$ is w-local and the set of closed points $X'_0 \subset X'$ is the inverse image of $X_0$. \medskip\noindent We obtain a continuous map $Y \to X'$ of underlying topological spaces over $X$ identifying $\pi_0(Y)$ with $\pi_0(X')$. By Lemma \ref{lemma-local-isomorphism-fully-faithful} (and Lemma \ref{lemma-ind-zariski-implies}) this is corresponds to a morphism of affine schemes $Y \to X'$ over $X$. Since $Y \to X$ maps $Y_0$ into $X_0$ we see that $Y \to X'$ maps $Y_0$ into $X'_0$, i.e., $Y \to X'$ is w-local. By Lemma \ref{lemma-w-local-morphism-equal-points-stalks-is-iso} we see that $Y \cong X'$ and we win.
The equivalence of (1) and (2) follows immediately from Proposition \ref{proposition-maps-wich-identify-local-rings}. \medskip\noindent Assume (3)(a) and (3)(b). Let $A \to B$ be faithfully flat and ind-Zariski. We will use without further mention the fact that a flat map $A \to B$ is faithfully flat if and only if every closed point of $\Spec(A)$ is in the image of $\Spec(B) \to \Spec(A)$ We will show that $A \to B$ has a section. \medskip\noindent Let $I \subset A$ be an ideal such that $V(I) \subset \Spec(A)$ is the set of closed points of $\Spec(A)$.  We may replace $B$ by the ring $C$ constructed in Lemma \ref{lemma-w-local-algebraic-residue-field-extensions} for $A \to B$ and $I \subset A$. Thus we may assume $\Spec(B)$ is w-local such that the set of closed points of $\Spec(B)$ is $V(IB)$. \medskip\noindent Assume $\Spec(B)$ is w-local and the set of closed points of $\Spec(B)$ is $V(IB)$. Choose a continuous section to the surjective continuous map $V(IB) \to V(I)$. This is possible as $V(I) \cong \pi_0(\Spec(A))$ is extremally disconnected, see Topology, Proposition \ref{topology-proposition-projective-in-category-hausdorff-qc}. The image is a closed subspace $T \subset \pi_0(\Spec(B)) \cong V(JB)$ mapping homeomorphically onto $\pi_0(A)$. Replacing $B$ by the ind-Zariski quotient ring constructed in Lemma \ref{lemma-construct} we see that we may assume $\pi_0(\Spec(B)) \to \pi_0(\Spec(A))$ is bijective. At this point $A \to B$ is an isomorphism by Lemma \ref{lemma-w-local-morphism-equal-points-stalks-is-iso}. \medskip\noindent Assume (1) or equivalently (2). Let $A \to A_w$ be the ring map constructed in Lemma \ref{lemma-make-w-local}. By (1) there is a section $A_w \to A$. Thus $\Spec(A)$ is homeomorphic to a closed subset of $\Spec(A_w)$. By Lemma \ref{lemma-closed-subspace-w-local} we see (3)(a) holds. Finally, let $T \to \pi_0(A)$ be a surjective map with $T$ an extremally disconnected, quasi-compact, Hausdorff topological space (Topology, Lemma \ref{topology-lemma-existence-projective-cover}). Choose $A \to B$ as in Lemma \ref{lemma-construct-profinite} adapted to $T \to \pi_0(\Spec(A))$. By (1) there is a section $B \to A$. Thus we see that $T = \pi_0(\Spec(B)) \to \pi_0(\Spec(A))$ has a section. A formal categorical argument, using Topology, Proposition \ref{topology-proposition-projective-in-category-hausdorff-qc}, implies that $\pi_0(\Spec(A))$ is extremally disconnected.
We first apply Lemma \ref{lemma-make-w-local} to see that we may assume that $\Spec(A)$ is w-local. Choose an extremally disconnected space $T$ and a surjective continuous map $T \to \pi_0(\Spec(A))$, see Topology, Lemma \ref{topology-lemma-existence-projective-cover}. Note that $T$ is profinite. Apply Lemma \ref{lemma-construct-profinite} to find an ind-Zariski ring map $A \to B$ such that $\pi_0(\Spec(B)) \to \pi_0(\Spec(A))$ realizes $T \to \pi_0(\Spec(A))$ and such that $$ \xymatrix{ \Spec(B) \ar[r] \ar[d] & \pi_0(\Spec(B)) \ar[d] \\ \Spec(A) \ar[r] & \pi_0(\Spec(A)) } $$ is cartesian in the category of topological spaces. Note that $\Spec(B)$ is w-local, that $\Spec(B) \to \Spec(A)$ is w-local, and that the set of closed points of $\Spec(B)$ is the inverse image of the set of closed points of $\Spec(A)$, see Lemma \ref{lemma-silly}. Thus condition (3) of Lemma \ref{lemma-w-local-extremally-disconnected} holds for $B$.
Let $\overline{C}$ be an ind-\'etale $A/I$-algebra. Consider the category $\mathcal{C}$ of factorizations $A \to B \to \overline{C}$ where $A \to B$ is \'etale. (We ignore some set theoretical issues in this proof.) We will show that this category is directed and that $C = \colim_\mathcal{C} B$ is an ind-\'etale $A$-algebra such that $\overline{C} = C/IC$. \medskip\noindent We first prove that $\mathcal{C}$ is directed (Categories, Definition \ref{categories-definition-directed}). The category is nonempty as $A \to A \to \overline{C}$ is an object. Suppose that $A \to B \to \overline{C}$ and $A \to B' \to \overline{C}$ are two objects of $\mathcal{C}$. Then $A \to B \otimes_A B' \to \overline{C}$ is another (use Algebra, Lemma \ref{algebra-lemma-etale}). Suppose that $f, g : B \to B'$ are two maps between objects $A \to B \to \overline{C}$ and $A \to B' \to \overline{C}$ of $\mathcal{C}$. Then a coequalizer is $A \to B' \otimes_{f, B, g} B' \to \overline{C}$. This is an object of $\mathcal{C}$ by Algebra, Lemmas \ref{algebra-lemma-etale} and \ref{algebra-lemma-map-between-etale}. Thus the category $\mathcal{C}$ is directed. \medskip\noindent Write $\overline{C} = \colim \overline{B_i}$ as a filtered colimit with $\overline{B_i}$ \'etale over $A/I$. For every $i$ there exists $A \to B_i$ \'etale with $\overline{B_i} = B_i/IB_i$, see Algebra, Lemma \ref{algebra-lemma-lift-etale}. Thus $C \to \overline{C}$ is surjective. Since $C/IC \to \overline{C}$ is ind-\'etale (Lemma \ref{lemma-ind-etale-permanence}) we see that it is flat. Hence $\overline{C}$ is a localization of $C/IC$ at some multiplicative subset $S \subset C/IC$ (Algebra, Lemma \ref{algebra-lemma-pure}). Take an $f \in C$ mapping to an element of $S \subset C/IC$. Choose $A \to B \to \overline{C}$ in $\mathcal{C}$ and $g \in B$ mapping to $f$ in the colimit. Then we see that $A \to B_g \to \overline{C}$ is an object of $\mathcal{C}$ as well. Thus $f$ is an invertible element of $C$. It follows that $C/IC = \overline{C}$. \medskip\noindent Next, we claim that for an ind-\'etale algebra $D$ over $A$ we have $$ \Mor_A(D, C) = \Mor_{A/I}(D/ID, \overline{C}) $$ Namely, let $D/ID \to \overline{C}$ be an $A/I$-algebra map. Write $D = \colim_{i \in I} D_i$ as a colimit over a directed set $I$ with $D_i$ \'etale over $A$. By choice of $\mathcal{C}$ we obtain a transformation $I \to \mathcal{C}$ and hence a map $D \to C$ compatible with maps to $\overline{C}$. Whence the claim. \medskip\noindent It follows that the functor $v$ defined by the rule $$ \overline{C}  \longmapsto v(\overline{C}) = \colim_{A \to B \to \overline{C}} B $$ is a right adjoint to the base change functor $u$ as required by the lemma. The functor $v$ is fully faithful because $u \circ v = \text{id}$ by construction, see Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
Set $T^1(A) = T(A)$ and $T^{n + 1}(A) = T(T^n(A))$. Let $$ C = \colim T^n(A) $$ This algebra is faithfully flat over each $T^n(A)$ and in particular over $A$, see Algebra, Lemma \ref{algebra-lemma-colimit-faithfully-flat}. Moreover, $C$ is ind-\'etale over $A$ by Lemma \ref{lemma-ind-ind-etale}. If $C \to B$ is \'etale, then there exists an $n$ and an \'etale ring map $T^n(A) \to B'$ such that $B = C \otimes_{T^n(A)} B'$, see Algebra, Lemma \ref{algebra-lemma-etale}. If $C \to B$ is faithfully flat, then $\Spec(B) \to \Spec(C) \to \Spec(T^n(A))$ is surjective, hence $\Spec(B') \to \Spec(T^n(A))$ is surjective. In other words, $T^n(A) \to B'$ is faithfully flat. By our construction, there is a $T^n(A)$-algebra map $B' \to T^{n + 1}(A)$. This induces a $C$-algebra map $B \to C$ which finishes the proof.
Let $\mathfrak m$ be a maximal ideal of $A$. Let $A \to B$ be an \'etale ring map and let $\mathfrak q \subset B$ be a prime lying over $\mathfrak m$. By the description of the strict henselization $A_\mathfrak m^{sh}$ in Algebra, Lemma \ref{algebra-lemma-strict-henselization-different} it suffices to show that $A_\mathfrak m = B_\mathfrak q$. Note that there are finitely many primes $\mathfrak q = \mathfrak q_1, \mathfrak q_2, \ldots, \mathfrak q_n$ lying over $\mathfrak m$ and there are no specializations between them as an \'etale ring map is quasi-finite, see Algebra, Lemma \ref{algebra-lemma-etale-quasi-finite}. Thus $\mathfrak q_i$ is a maximal ideal and we can find $g \in \mathfrak q_2 \cap \ldots \cap \mathfrak q_n$, $g \not \in \mathfrak q$ (Algebra, Lemma \ref{algebra-lemma-silly}). After replacing $B$ by $B_g$ we see that $\mathfrak q$ is the only prime of $B$ lying over $\mathfrak m$. The image $U \subset \Spec(A)$ of $\Spec(B) \to \Spec(A)$ is open (Algebra, Proposition \ref{algebra-proposition-fppf-open}). Thus the complement $\Spec(A) \setminus U$ is closed and we can find $f \in A$, $f \not \in \mathfrak p$ such that $\Spec(A) = U \cup D(f)$. The ring map $A \to B \times A_f$ is faithfully flat and \'etale, hence has a section $\sigma : B \times A_f \to A$ by assumption on $A$. Observe that $\sigma$ is \'etale, hence flat as a map between \'etale $A$-algebras (Algebra, Lemma \ref{algebra-lemma-map-between-etale}). Since $\mathfrak q$ is the only prime of $B \times A_f$ lying over $A$ we find that $A_\mathfrak p \to B_\mathfrak q$ has a section which is also flat. Thus $A_\mathfrak p \to B_\mathfrak q \to A_\mathfrak p$ are flat local ring maps whose composition is the identity. Since a flat local homomorphism of local rings is injective we conclude these maps are isomorphisms as desired.
There is a faithfully flat, ind-Zariski ring map $A \to A'$ such that $\Spec(A')$ is w-local and such that the set of closed points of $\Spec(A')$ maps onto $\Spec(A)$, see Lemma \ref{lemma-make-w-local}. Let $I \subset A'$ be the ideal such that $V(I)$ is the set of closed points of $\Spec(A')$. Choose $A' \to C'$ as in Lemma \ref{lemma-first-construction}. Note that the local rings $C'_{\mathfrak m'}$ at maximal ideals $\mathfrak m' \subset C'$ are strictly henselian by Lemma \ref{lemma-have-sections-strictly-henselian}. We apply Lemma \ref{lemma-w-local-algebraic-residue-field-extensions} to $A' \to C'$ and $I \subset A'$ to get $C' \to C$ with $C'/IC' \cong C/IC$. Note that since $A' \to C'$ is faithfully flat, $\Spec(C'/IC')$ surjects onto the set of closed points of $A'$ and in particular onto $\Spec(A)$. Moreover, as $V(IC) \subset \Spec(C)$ is the set of closed points of $C$ and $C' \to C$ is ind-Zariski (and identifies local rings) we obtain properties (1), (3), (7), and (9). \medskip\noindent Denote $J \subset C$ the ideal such that $V(J)$ is the set of closed points of $\Spec(C)$. Set $D' = B \otimes_A C$. The ring map $C \to D'$ induces algebraic residue field extensions. Keep in mind that since $V(J) \to \Spec(A)$ is surjective the map $T = V(JD) \to \Spec(B)$ is surjective too. Apply Lemma \ref{lemma-w-local-algebraic-residue-field-extensions} to $C \to D'$ and $J \subset C$ to get  $D' \to D$ with $D'/JD' \cong D/JD$. All of the remaining properties given in the lemma are immediate from the results of Lemma \ref{lemma-w-local-algebraic-residue-field-extensions}.
A singleton $\{X \to Y\}$ is a standard V covering if and only if given a morphism $g : \Spec(V) \to Y$ there is an extension of valuation rings $V \subset W$ and a commutative diagram $$ \xymatrix{ \Spec(W) \ar[r] \ar[d] & X \ar[d] \\ \Spec(V) \ar[r]^g & Y } $$ Thus (1) $\Rightarrow$ (2) is immediate from the definition. Conversely, assume (2) and let $g : \Spec(V) \to Y$ as above be given. Write $\Spec(V) \times_Y X_i = \Spec(A_i)$. Since $\{X_i \to Y\}$ is a standard V covering, we may choose a valuation ring $W_i$ and a ring map $A_i \to W_i$ such that the composition $V \to A_i \to W_i$ is an extension of valuation rings. In particular, the quotient $A'_i$ of $A_i$ by its $V$-torsion is a faitfhully flat $V$-algebra. Flatness by More on Algebra, Lemma \ref{more-algebra-lemma-valuation-ring-torsion-free-flat} and surjectivity on spectra because $A_i \to W_i$ factors through $A'_i$. Thus $$ A = \colim A'_i $$ is a faithfully flat $V$-algebra (Algebra, Lemma \ref{algebra-lemma-colimit-faithfully-flat}). Since $\{\Spec(A) \to \Spec(V)\}$ is a standard fpqc cover, it is a standard V cover (Topologies, Lemma \ref{topologies-lemma-standard-fpqc-standard-V}) and hence we can choose $\Spec(W) \to \Spec(A)$ such that $V \to W$ is an extension of valuation rings. Since we can compose with the morphism $\Spec(A) \to X = \Spec(\colim A_i)$ the proof is complete.
Proof of (2) $\Rightarrow$ (1). Recall that a V covering given by a single arrow between affines is a standard V covering, see Topologies, Definition \ref{topologies-definition-V-covering} and Lemma \ref{topologies-lemma-refine-standard-V}. Recall that any ph covering is a V covering, see Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc-ph-V}. Hence if $X = \lim X_i$ as in (2), then $\{X_i \to Y\}$ is a standard V covering for each $i$. Thus by Lemma \ref{lemma-pro-V-V} we see that (1) is true. \medskip\noindent Proof of (3) $\Rightarrow$ (2). This is clear because an h covering is always a ph covering, see More on Flatness, Definition \ref{flat-definition-h-covering}. \medskip\noindent Proof of (1) $\Rightarrow$ (3). This is the interesting direction, but the interesting content in this proof is hidden in More on Flatness, Lemma \ref{flat-lemma-equivalence-h-v-locally-finite-presentation}. Write $X = \Spec(A)$ and $Y = \Spec(R)$. We can write $A = \colim A_i$ with $A_i$ of finite presentation over $R$, see Algebra, Lemma \ref{algebra-lemma-ring-colimit-fp}. Set $X_i = \Spec(A_i)$. Then $\{X_i \to Y\}$ is a standard V covering for all $i$ by (1) and Topologies, Lemma \ref{topologies-lemma-refine-standard-V}. Hence $\{X_i \to Y\}$ is an h covering by More on Flatness, Definition \ref{flat-definition-h-covering}. This finishes the proof.
We will prove this by verifying (1) and (2') of Topologies, Lemma \ref{topologies-lemma-sheaf-property-V}. The sheaf property for Zariski coverings follows from the fact that $F$ has the sheaf property for all h coverings. Finally, suppose that $X \to Y$ is a morphism of affine schemes over $S$ such that $\{X \to Y\}$ is a V covering. By Lemma \ref{lemma-pro-h-V} we can write $X = \lim X_i$ as a directed limit of affine schemes over $Y$ such that $\{X_i \to Y\}$ is an h covering for each $i$. We obtain \begin{align*} & \text{Equalizer}( \xymatrix{ F(X) \ar@<1ex>[r] \ar@<-1ex>[r] & F(X \times_Y X) } ) \\ & = \text{Equalizer}( \xymatrix{ \colim F(X_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \colim F(X_i \times_Y X_i) } ) \\ & = \colim \text{Equalizer}( \xymatrix{ F(X_i) \ar@<1ex>[r] \ar@<-1ex>[r] & F(X_i \times_Y X_i) } ) \\ & = \colim F(Y) = F(Y) \end{align*} which is what we wanted to show. The first equality because $F$ is limit preserving and $X = \lim X_i$ and $X \times_Y X = \lim X_i \times_Y X_i$. The second equality because filtered colimits are exact. The third equality because $F$ satisfies the sheaf property for h coverings.
The equivalence of (1) and (2) follows immediately from Proposition \ref{proposition-weakly-etale}. \medskip\noindent Assume (3)(a), (3)(b), and (3)(c). Let $A \to B$ be faithfully flat and ind-\'etale. We will use without further mention the fact that a flat map $A \to B$ is faithfully flat if and only if every closed point of $\Spec(A)$ is in the image of $\Spec(B) \to \Spec(A)$ We will show that $A \to B$ has a section. \medskip\noindent Let $I \subset A$ be an ideal such that $V(I) \subset \Spec(A)$ is the set of closed points of $\Spec(A)$.  We may replace $B$ by the ring $C$ constructed in Lemma \ref{lemma-w-local-algebraic-residue-field-extensions} for $A \to B$ and $I \subset A$. Thus we may assume $\Spec(B)$ is w-local such that the set of closed points of $\Spec(B)$ is $V(IB)$. In this case $A \to B$ identifies local rings by condition (3)(c) as it suffices to check this at maximal ideals of $B$ which lie over maximal ideals of $A$. Thus $A \to B$ has a section by Lemma \ref{lemma-w-local-extremally-disconnected}. \medskip\noindent Assume (1) or equivalently (2). We have (3)(c) by Lemma \ref{lemma-have-sections-strictly-henselian}. Properties (3)(a) and (3)(b) follow from Lemma \ref{lemma-w-local-extremally-disconnected}.
Set $X = \Spec(A)$ and $Y = \Spec(B)$. Choose a finite partition $X = \coprod X_i$ and $X'_i \to X_i$ as in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-decompose-quasi-finite-morphism}. Because $X$ is extremally disconnected, every constructible locally closed subset is open and closed, hence we see that $X$ is topologically the disjoint union of the strata $X_i$. Thus we may replace $X$ by the $X_i$ and assume there exists a surjective finite locally free morphism $X' \to X$ such that $(X' \times_X Y)_{red}$ is isomorphic to a finite disjoint union of copies of $X'_{red}$. Picture $$ \xymatrix{ \coprod_{i = 1, \ldots, r} X' \ar[r] \ar[d] & Y \ar[d] \\ X' \ar[r] & X } $$ The assumption on the residue fields of $A$ implies that this diagram is a fibre product diagram on underlying sets of points (details omitted). Since $X$ is extremally disconnected and $X'$ is Hausdorff (Lemma \ref{lemma-profinite-goes-up}), the continuous map $X' \to X$ has a continuous section $\sigma$. Then $\coprod_{i = 1, \ldots, r} \sigma(X) \to Y$ is a bijective continuous map. By Topology, Lemma \ref{topology-lemma-bijective-map} we see that it is a homeomorphism and the proof is done.
We will use the criterion of Lemma \ref{lemma-w-local-strictly-henselian-extremally-disconnected}. Set $X = \Spec(A)$ and $Y = \Spec(B)$. As $Y \to X$ is a finite morphism, we see that the set of closed points $Y_0$ of $Y$ is the inverse image of the set of closed points $X_0$ of $X$. Moreover, every point of $Y$ specializes to a unique point of $Y_0$ as (a) this is true for $X$ and (b) the map $X \to Y$ is separated. For every $y \in Y_0$ with image $x \in X_0$ we see that $\mathcal{O}_{Y, y}$ is strictly henselian by Algebra, Lemma \ref{algebra-lemma-finite-over-henselian} applied to $\mathcal{O}_{X, x} \to B \otimes_A \mathcal{O}_{X, x}$. It remains to show that $Y_0$ is extremally disconnected. To do this we look at $X_0 \times_X Y \to X_0$ where $X_0 \subset X$ is the reduced induced scheme structure. Note that the underlying topological space of $X_0 \times_X Y$ agrees with $Y_0$. Now the desired result follows from Lemma \ref{lemma-finite-finitely-presented-over-extremally-disconnected}.
This follows from the corresponding result for fpqc coverings (Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}), Lemma \ref{lemma-recognize-proetale-covering}, and the fact that an \'etale morphism is a weakly \'etale morphism, see More on Morphisms, Lemma \ref{more-morphisms-lemma-when-weakly-etale}.
This follows from the fact that composition and base changes of weakly \'etale morphisms are weakly \'etale (More on Morphisms, Lemmas \ref{more-morphisms-lemma-composition-weakly-etale} and \ref{more-morphisms-lemma-base-change-weakly-etale}), Lemma \ref{lemma-recognize-proetale-covering}, and the corresponding results for fpqc coverings, see Topologies, Lemma \ref{topologies-lemma-fpqc}.
Let us show that $S_\proetale$ is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition \ref{sites-definition-site}. Since $(\Sch/S)_\proetale$ is a site, it suffices to prove that given any covering $\{U_i \to U\}$ of $(\Sch/S)_\proetale$ with $U \in \Ob(S_\proetale)$ we also have $U_i \in \Ob(S_\proetale)$. This follows from the definitions as the composition of weakly \'etale morphisms is weakly \'etale. \medskip\noindent To show that $(\textit{Aff}/S)_\proetale$ is a site, reasoning as above, it suffices to show that the collection of standard pro-\'etale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition \ref{sites-definition-site}. This follows from Lemma \ref{lemma-recognize-proetale-covering} and the corresponding result for standard fpqc coverings (Topologies, Lemma \ref{topologies-lemma-fpqc-affine-axioms}).
The category $\Sch_\proetale$ contains $\Spec(\mathbf{Z})$ and is closed under products and fibre products 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 $U, V, W \in \Ob(\Sch_\proetale)$. The fibre product $V \times_U W$ in $\Sch_\proetale$ 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 $(\Sch/S)_\proetale$. This proves the result for $(\Sch/S)_\proetale$. If $U \to S$, $V \to U$ and $W \to U$ are weakly \'etale then so is $V \times_U W \to S$ (see More on Morphisms, Section \ref{more-morphisms-section-weakly-etale}) and hence we get fibre products for $S_\proetale$. If $U, V, W$ are affine, so is $V \times_U W$ and hence we get fibre products for $(\textit{Aff}/S)_\proetale$. \medskip\noindent Let $a, b : U \to V$ be two morphisms in $\Sch_\proetale$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is $$ V \times_{\Delta_{V/\Spec(\mathbf{Z})}, V \times_{\Spec(\mathbf{Z})} V, (a, b)} (U \times_{\Spec(\mathbf{Z})} U) $$ which is an object of $\Sch_\proetale$ by what we saw above. Thus $\Sch_\proetale$ has equalizers. If $a$ and $b$ are morphisms over $S$, then the equalizer (in the category of schemes) is also given by $$ V \times_{\Delta_{V/S}, V \times_S V, (a, b)} (U \times_S U) $$ hence we see that $(\Sch/S)_\proetale$ has equalizers. Moreover, if $U$ and $V$ are weakly-\'etale over $S$, then so is the equalizer above as a fibre product of schemes weakly \'etale over $S$. Thus $S_\proetale$ has equalizers. The statements on final objects is clear.
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 $u : (\textit{Aff}/S)_\proetale \to (\Sch/S)_\proetale$. Being cocontinuous just means that any pro-\'etale covering of $T/S$, $T$ affine, can be refined by a standard pro-\'etale covering of $T$. This is the content of Lemma \ref{lemma-proetale-affine}. Hence (1) holds. We see $u$ is continuous simply because a standard pro-\'etale covering is a pro-\'etale 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.