hcju/stacksqa huggingface dataset:
Let us show that $S_{Zar}$ 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 $(\Sch/S)_{Zar}$ is a site, it suffices to prove that given any covering $\{U_i \to U\}$ of $(\Sch/S)_{Zar}$ with $U \in \Ob(S_{Zar})$ we also have $U_i \in \Ob(S_{Zar})$. This follows from the definitions as the composition of open immersions is an open immersion. \medskip\noindent Let us show that $(\textit{Aff}/S)_{Zar}$ is a site. Reasoning as above, it suffices to show that the collection of standard Zariski coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition \ref{sites-definition-site}. Let $R$ be a ring. Let $f_1, \ldots, f_n \in R$ generate the unit ideal. For each $i \in \{1, \ldots, n\}$ let $g_{i1}, \ldots, g_{in_i} \in R_{f_i}$ be elements generating the unit ideal of $R_{f_i}$. Write $g_{ij} = f_{ij}/f_i^{e_{ij}}$ which is possible. After replacing $f_{ij}$ by $f_i f_{ij}$ if necessary, we have that $D(f_{ij}) \subset D(f_i) \cong \Spec(R_{f_i})$ is equal to $D(g_{ij}) \subset \Spec(R_{f_i})$. Hence we see that the family of morphisms $\{D(g_{ij}) \to \Spec(R)\}$ is a standard Zariski covering. From these considerations it follows that (2) holds for standard Zariski coverings. We omit the verification of (1) and (3).
For $\Sch_{Zar}$ it is true by construction, see Sets, Lemma \ref{sets-lemma-what-is-in-it}. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \Ob(\Sch_{Zar})$. The fibre product $V \times_U W$ in $\Sch_{Zar}$ 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 $(\Sch/S)_{Zar}$. This proves the result for $(\Sch/S)_{Zar}$. If $U \to S$, $V \to U$ and $W \to U$ are open immersions then so is $V \times_U W \to S$ and hence we get the result for $S_{Zar}$. If $U, V, W$ are affine, so is $V \times_U W$ and hence the result for $(\textit{Aff}/S)_{Zar}$.
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 $u : (\textit{Aff}/S)_{Zar} \to (\Sch/S)_{Zar}$. Being cocontinuous just means that any Zariski covering of $T/S$, $T$ affine, can be refined by a standard Zariski covering of $T$. This is the content of Lemma \ref{lemma-zariski-affine}. Hence (1) holds. We see $u$ is continuous simply because a standard Zariski covering is a Zariski 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.
We will use repeatedly that for any object $U/S$ of $S_{Zar}$ the morphism $U \to S$ is an isomorphism onto an open subscheme. Let $\mathcal{F}$ be a sheaf on $S$. Then we define a sheaf on $S_{Zar}$ by the rule $\mathcal{F}'(U/S) = \mathcal{F}(\Im(U \to S))$. For the converse, we choose for every open subscheme $U \subset S$ an object $U'/S \in \Ob(S_{Zar})$ with $\Im(U' \to S) = U$ (here you have to use Sets, Lemma \ref{sets-lemma-what-is-in-it}). Given a sheaf $\mathcal{G}$ on $S_{Zar}$ we define a sheaf on $S$ by setting $\mathcal{G}'(U) = \mathcal{G}(U'/S)$. To see that $\mathcal{G}'$ is a sheaf we use that for any open covering $U = \bigcup_{i \in I} U_i$ the covering $\{U_i \to U\}_{i \in I}$ is combinatorially equivalent to a covering $\{U_j' \to U'\}_{j \in J}$ in $S_{Zar}$ by Sets, Lemma \ref{sets-lemma-coverings-site}, and we use Sites, Lemma \ref{sites-lemma-tautological-same-sheaf}. Details omitted.
Denote the functor $u : T_{Zar} \to (\Sch/S)_{Zar}$. In other words, given and open immersion $j : U \to T$ corresponding to an object of $T_{Zar}$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma \ref{lemma-fibre-products-Zariski}. Moreover, $T_{Zar}$ has equalizers (as any two morphisms with the same source and target are the same) and $u$ commutes with them. It is clearly cocontinuous. It is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the lemma follows from Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-preserve-equalizers}.
Since the diagram is cartesian, we have for $U'/S'$ that $U' \times_{S'} T' = U' \times_S T$. Hence both $i_g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\Sch/T)_{Zar}$ to the sheaf $U' \mapsto \mathcal{F}(U' \times_{S'} T')$ on $S'_{Zar}$ (use Lemmas \ref{lemma-put-in-T} and \ref{lemma-morphism-big-small}). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma \ref{sites-lemma-localize-morphism}.
Given a sheaf $\mathcal{F}$ on $\Sh((\Sch/S)_{Zar})$ we set $\mathcal{F}_T = i_p^{-1}\mathcal{F}$ where $p : T \to S$ is the structure morphism. Note that $\mathcal{F}_T(U) = \mathcal{F}(U'/S)$ for any open $U \subset T$, and $U' \to T$ an open immersion in $(\Sch/T)_{Zar}$ with image $U$, see Lemmas \ref{lemma-Zariski-usual} and \ref{lemma-put-in-T}. Hence given $f : T' \to T$ over $S$ and $U, U' \to T$ we get a canonical map $\mathcal{F}_T(U) = \mathcal{F}(U'/S) \to \mathcal{F}(U'\times_T T'/S) = \mathcal{F}_{T'}(f^{-1}(U))$ where the middle is the restriction map of $\mathcal{F}$ with respect to the morphism $U' \times_T T' \to U'$ over $S$. The collection of these maps are compatible with restrictions, and hence define an $f$-map $c_f$ from $\mathcal{F}_T$ to $\mathcal{F}_{T'}$, see Sheaves, Definition \ref{sheaves-definition-f-map} and the discussion surrounding it. It is clear that $c_{f \circ g}$ is the composition of $c_f$ and $c_g$, since composition of restriction maps of $\mathcal{F}$ gives restriction maps. \medskip\noindent Conversely, given a system $(\mathcal{F}_T, c_f)$ as in the lemma we may define a presheaf $\mathcal{F}$ on $\Sh((\Sch/S)_{Zar})$ by simply setting $\mathcal{F}(T/S) = \mathcal{F}_T(T)$. As restriction mapping, given $f : T' \to T$ we set for $s \in \mathcal{F}(T)$ the pullback $f^*(s)$ equal to $c_f(s)$ (where we think of $c_f$ as an $f$-map again). The condition on the $c_f$ guarantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse.
For each $i$ choose an affine open covering $T_i = \bigcup_{j \in J_i} T_{ij}$ such that each $T_{ij}$ maps into an affine open subscheme of $T$. By Lemma \ref{lemma-etale} the refinement $\{T_{ij} \to T\}_{i \in I, j \in J_i}$ is an \'etale covering of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into an affine open $W_i$ of $T$. Applying Sets, Lemma \ref{sets-lemma-what-is-in-it} we see that $W_i$ is isomorphic to an object of $\Sch_\etale$. But then $T_i$ as a finite type scheme over $W_i$ is isomorphic to an object $V_i$ of $\Sch_\etale$ by a second application of Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{V_i \to T\}_{i \in I}$ refines $\{T_i \to T\}_{i \in I}$ (because they are isomorphic). Moreover, $\{V_i \to T\}_{i \in I}$ is combinatorially equivalent to a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_\etale$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{U_j \to T\}_{j \in J}$ is a refinement as in (1). In the situation of (2), (3) each of the schemes $T_i$ is isomorphic to an object of $\Sch_\etale$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}, and another application of Sets, Lemma \ref{sets-lemma-coverings-site} gives what we want.
Let us show that $S_\etale$ 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 $(\Sch/S)_\etale$ is a site, it suffices to prove that given any covering $\{U_i \to U\}$ of $(\Sch/S)_\etale$ with $U \in \Ob(S_\etale)$ we also have $U_i \in \Ob(S_\etale)$. This follows from the definitions as the composition of \'etale morphisms is an \'etale morphism. \medskip\noindent Let us show that $(\textit{Aff}/S)_\etale$ is a site. Reasoning as above, it suffices to show that the collection of standard \'etale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition \ref{sites-definition-site}. This is clear since for example, given a standard \'etale covering $\{T_i \to T\}_{i\in I}$ and for each $i$ we have a standard \'etale covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a standard \'etale covering because $\bigcup_{i\in I} J_i$ is finite and each $T_{ij}$ is affine.
For $\Sch_\etale$ it is true by construction, see Sets, Lemma \ref{sets-lemma-what-is-in-it}. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \Ob(\Sch_\etale)$. The fibre product $V \times_U W$ in $\Sch_\etale$ 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 $(\Sch/S)_\etale$. This proves the result for $(\Sch/S)_\etale$. If $U \to S$, $V \to U$ and $W \to U$ are \'etale then so is $V \times_U W \to S$ and hence we get the result for $S_\etale$. If $U, V, W$ are affine, so is $V \times_U W$ and hence the result for $(\textit{Aff}/S)_\etale$.
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 $u : (\textit{Aff}/S)_\etale \to (\Sch/S)_\etale$. Being cocontinuous just means that any \'etale covering of $T/S$, $T$ affine, can be refined by a standard \'etale covering of $T$. This is the content of Lemma \ref{lemma-etale-affine}. Hence (1) holds. We see $u$ is continuous simply because a standard \'etale covering is a \'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.
Denote the functor $u : T_\etale \to (\Sch/S)_\etale$. In other words, given an \'etale morphism $j : U \to T$ corresponding to an object of $T_\etale$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma \ref{lemma-fibre-products-etale}. Let $a, b : U \to V$ be two morphisms in $T_\etale$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is $$ V \times_{\Delta_{V/T}, V \times_T V, (a, b)} U \times_T U $$ which is a fibre product of schemes \'etale over $T$, hence \'etale over $T$. Thus $T_\etale$ has equalizers and $u$ commutes with them. It is clearly cocontinuous. It is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the Lemma follows from Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-preserve-equalizers}.
The functor $u$ is cocontinuous, continuous and commutes with fibre products and equalizers (details omitted; compare with the proof of Lemma \ref{lemma-put-in-T-etale}). Hence Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-preserve-equalizers} apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/T$ and $V/S$ we have $\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)$ as desired. Thus we may apply Sites, Lemmas \ref{sites-lemma-have-functor-other-way} and \ref{sites-lemma-have-functor-other-way-morphism} to get the formula for $f_{big, *}$.
The equality $i_f = f_{big} \circ i_T$ follows from the equality $i_f^{-1} = i_T^{-1} \circ f_{big}^{-1}$ which is clear from the descriptions of these functors above. Thus we see (1). \medskip\noindent The functor $u : S_\etale \to T_\etale$, $u(U \to S) = (U \times_S T \to T)$ transforms coverings into coverings and commutes with fibre products, see Lemma \ref{lemma-etale} (3) and \ref{lemma-fibre-products-etale}. Moreover, both $S_\etale$, $T_\etale$ have final objects, namely $S/S$ and $T/T$ and $u(S/S) = T/T$. Hence by Sites, Proposition \ref{sites-proposition-get-morphism} the functor $u$ corresponds to a morphism of sites $T_\etale \to S_\etale$. This in turn gives rise to the morphism of topoi, see Sites, Lemma \ref{sites-lemma-morphism-sites-topoi}. The description of the pushforward is clear from these references. \medskip\noindent Part (3) follows because $\pi_S$ and $\pi_T$ are given by the inclusion functors and $f_{small}$ and $f_{big}$ by the base change functors $U \mapsto U \times_S T$. \medskip\noindent Statement (4) follows from (3) by precomposing with $i_T$.
Since the diagram is cartesian, we have for $U'/S'$ that $U' \times_{S'} T' = U' \times_S T$. Hence both $i_g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\Sch/T)_\etale$ to the sheaf $U' \mapsto \mathcal{F}(U' \times_{S'} T')$ on $S'_\etale$ (use Lemmas \ref{lemma-put-in-T-etale} and \ref{lemma-morphism-big-etale}). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma \ref{sites-lemma-localize-morphism}.
Given a sheaf $\mathcal{F}$ on $\Sh((\Sch/S)_\etale)$ we set $\mathcal{F}_T = i_p^{-1}\mathcal{F}$ where $p : T \to S$ is the structure morphism. Note that $\mathcal{F}_T(U) = \mathcal{F}(U/S)$ for any $U \to T$ in $T_\etale$ see Lemma \ref{lemma-put-in-T-etale}. Hence given $f : T' \to T$ over $S$ and $U \to T$ we get a canonical map $\mathcal{F}_T(U) = \mathcal{F}(U/S) \to \mathcal{F}(U \times_T T'/S) = \mathcal{F}_{T'}(U \times_T T')$ where the middle is the restriction map of $\mathcal{F}$ with respect to the morphism $U \times_T T' \to U$ over $S$. The collection of these maps are compatible with restrictions, and hence define a map $c'_f : \mathcal{F}_T \to f_{small, *}\mathcal{F}_{T'}$ where $u : T_\etale \to T'_\etale$ is the base change functor associated to $f$. By adjunction of $f_{small, *}$ (see Sites, Section \ref{sites-section-continuous-functors}) with $f_{small}^{-1}$ this is the same as a map $c_f : f_{small}^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}$. It is clear that $c'_{f \circ g}$ is the composition of $c'_f$ and $f_{small, *}c'_g$, since composition of restriction maps of $\mathcal{F}$ gives restriction maps, and this gives the desired relationship among $c_f$, $c_g$ and $c_{f \circ g}$. \medskip\noindent Conversely, given a system $(\mathcal{F}_T, c_f)$ as in the lemma we may define a presheaf $\mathcal{F}$ on $\Sh((\Sch/S)_\etale)$ by simply setting $\mathcal{F}(T/S) = \mathcal{F}_T(T)$. As restriction mapping, given $f : T' \to T$ we set for $s \in \mathcal{F}(T)$ the pullback $f^*(s)$ equal to $c_f(s)$ where we think of $c_f$ as a map $\mathcal{F}_T \to f_{small, *}\mathcal{F}_{T'}$ again. The condition on the $c_f$ guarantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse.
For each $i$ choose an affine open covering $T_i = \bigcup_{j \in J_i} T_{ij}$ such that each $T_{ij}$ maps into an affine open subscheme of $T$. By Lemma \ref{lemma-smooth} the refinement $\{T_{ij} \to T\}_{i \in I, j \in J_i}$ is a smooth covering of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into an affine open $W_i$ of $T$. Applying Sets, Lemma \ref{sets-lemma-what-is-in-it} we see that $W_i$ is isomorphic to an object of $\Sch_{smooth}$. But then $T_i$ as a finite type scheme over $W_i$ is isomorphic to an object $V_i$ of $\Sch_{smooth}$ by a second application of Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{V_i \to T\}_{i \in I}$ refines $\{T_i \to T\}_{i \in I}$ (because they are isomorphic). Moreover, $\{V_i \to T\}_{i \in I}$ is combinatorially equivalent to a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_{smooth}$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{U_j \to T\}_{j \in J}$ is a refinement as in (1). In the situation of (2), (3) each of the schemes $T_i$ is isomorphic to an object of $\Sch_{smooth}$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}, and another application of Sets, Lemma \ref{sets-lemma-coverings-site} gives what we want.
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 $u : (\textit{Aff}/S)_{smooth} \to (\Sch/S)_{smooth}$. Being cocontinuous just means that any smooth covering of $T/S$, $T$ affine, can be refined by a standard smooth covering of $T$. This is the content of Lemma \ref{lemma-smooth-affine}. Hence (1) holds. We see $u$ is continuous simply because a standard smooth covering is a smooth 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.
For each $i$ choose an affine open covering $T_i = \bigcup_{j \in J_i} T_{ij}$ such that each $T_{ij}$ maps into an affine open subscheme of $T$. By Lemma \ref{lemma-syntomic} the refinement $\{T_{ij} \to T\}_{i \in I, j \in J_i}$ is a syntomic covering of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into an affine open $W_i$ of $T$. Applying Sets, Lemma \ref{sets-lemma-what-is-in-it} we see that $W_i$ is isomorphic to an object of $\Sch_{syntomic}$. But then $T_i$ as a finite type scheme over $W_i$ is isomorphic to an object $V_i$ of $\Sch_{syntomic}$ by a second application of Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{V_i \to T\}_{i \in I}$ refines $\{T_i \to T\}_{i \in I}$ (because they are isomorphic). Moreover, $\{V_i \to T\}_{i \in I}$ is combinatorially equivalent to a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_{syntomic}$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{U_j \to T\}_{j \in J}$ is a covering as in (1). In the situation of (2), (3) each of the schemes $T_i$ is isomorphic to an object of $\Sch_{syntomic}$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}, and another application of Sets, Lemma \ref{sets-lemma-coverings-site} gives what we want.
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 $u : (\textit{Aff}/S)_{syntomic} \to (\Sch/S)_{syntomic}$. Being cocontinuous just means that any syntomic covering of $T/S$, $T$ affine, can be refined by a standard syntomic covering of $T$. This is the content of Lemma \ref{lemma-syntomic-affine}. Hence (1) holds. We see $u$ is continuous simply because a standard syntomic covering is a syntomic 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.
The first assertion is clear. The second follows as the composition of flat morphisms is flat (see Morphisms, Lemma \ref{morphisms-lemma-composition-flat}) and the composition of morphisms of finite presentation is of finite presentation (see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-presentation}). The third follows as the base change of a flat morphism is flat (see Morphisms, Lemma \ref{morphisms-lemma-base-change-flat}) and the base change of a morphism of finite presentation is of finite presentation (see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-presentation}). Moreover, the base change of a surjective family of morphisms is surjective (proof omitted).
For each $i$ choose an affine open covering $T_i = \bigcup_{j \in J_i} T_{ij}$ such that each $T_{ij}$ maps into an affine open subscheme of $T$. By Lemma \ref{lemma-fppf} the refinement $\{T_{ij} \to T\}_{i \in I, j \in J_i}$ is an fppf covering of $T$ as well. Hence we may assume each $T_i$ is affine, and maps into an affine open $W_i$ of $T$. Applying Sets, Lemma \ref{sets-lemma-what-is-in-it} we see that $W_i$ is isomorphic to an object of $\Sch_{fppf}$. But then $T_i$ as a finite type scheme over $W_i$ is isomorphic to an object $V_i$ of $\Sch_{fppf}$ by a second application of Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{V_i \to T\}_{i \in I}$ refines $\{T_i \to T\}_{i \in I}$ (because they are isomorphic). Moreover, $\{V_i \to T\}_{i \in I}$ is combinatorially equivalent to a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_{fppf}$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}. The covering $\{U_j \to T\}_{j \in J}$ is a refinement as in (1). In the situation of (2), (3) each of the schemes $T_i$ is isomorphic to an object of $\Sch_{fppf}$ by Sets, Lemma \ref{sets-lemma-what-is-in-it}, and another application of Sets, Lemma \ref{sets-lemma-coverings-site} gives what we want.
Let us show that $(\textit{Aff}/S)_{fppf}$ is a site. Reasoning as in the proof of Lemma \ref{lemma-verify-site-etale} it suffices to show that the collection of standard fppf coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition \ref{sites-definition-site}. This is clear since for example, given a standard fppf covering $\{T_i \to T\}_{i\in I}$ and for each $i$ we have a standard fppf covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a standard fppf covering because $\bigcup_{i\in I} J_i$ is finite and each $T_{ij}$ is affine.
For $\Sch_{fppf}$ it is true by construction, see Sets, Lemma \ref{sets-lemma-what-is-in-it}. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \Ob(\Sch_{fppf})$. The fibre product $V \times_U W$ in $\Sch_{fppf}$ 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 $(\Sch/S)_{fppf}$. This proves the result for $(\Sch/S)_{fppf}$. If $U, V, W$ are affine, so is $V \times_U W$ and hence the result for $(\textit{Aff}/S)_{fppf}$.
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 $u : (\textit{Aff}/S)_{fppf} \to (\Sch/S)_{fppf}$. Being cocontinuous just means that any fppf covering of $T/S$, $T$ affine, can be refined by a standard fppf covering of $T$. This is the content of Lemma \ref{lemma-fppf-affine}. Hence (1) holds. We see $u$ is continuous simply because a standard fppf covering is a fppf 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.
Let $f : U \to T$ be proper surjective and let an affine open covering $U = \bigcup_{j = 1, \ldots, m} U_j$ be given as in Definition \ref{definition-standard-ph-covering}. Then $U \times_T T' \to T'$ is proper surjective (Morphisms, Lemmas \ref{morphisms-lemma-base-change-surjective} and \ref{morphisms-lemma-base-change-proper}). Also, $U \times_T T' = \bigcup_{j = 1, \ldots, m} U_j \times_T T'$ is an affine open covering. This concludes the proof.