hcju/stacksqa huggingface dataset:
Proof of (1). Let $\mathcal{F}$ be a sheaf on $T_\etale$. Let $g : S' \to S$ be an object of $(\Sch/S)_\etale$. Consider the fibre product $$ \xymatrix{ T' \ar[r]_{f'} \ar[d]_{g'} & S' \ar[d]^g \\ T \ar[r]^f & S } $$ Then we have $$ (f_{big, *}\pi_T^{-1}\mathcal{F})(S') = (\pi_T^{-1}\mathcal{F})(T') = ((g'_{small})^{-1}\mathcal{F})(T')  = (f'_{small, *}(g'_{small})^{-1}\mathcal{F})(S') $$ the second equality by Lemma \ref{lemma-describe-pullback}. On the other hand $$ (\pi_S^{-1}f_{small, *}\mathcal{F})(S') = (g_{small}^{-1}f_{small, *}\mathcal{F})(S') $$ again by Lemma \ref{lemma-describe-pullback}. Hence by proper base change for sheaves of sets (Lemma \ref{lemma-proper-base-change-f-star}) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism $\pi_S^{-1}f_{small, *}\mathcal{F} = f_{big, *}\pi_T^{-1}\mathcal{F}$. Thus an isomorphism of functors $\pi_S^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_T^{-1}$. \medskip\noindent Proof of (2). There is a canonical base change map $\pi_S^{-1}Rf_{small, *}K \to Rf_{big, *}\pi_T^{-1}K$ for any $K$ in $D(T_\etale)$, see Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}. To prove it is an isomorphism, it suffices to prove the pull back of the base change map by $i_g : \Sh(S'_\etale) \to \Sh((\Sch/S)_\etale)$ is an isomorphism for any object $g : S' \to S$ of $(\Sch/S)_\etale$. Let $T', g', f'$ be as in the previous paragraph. The pullback of the base change map is \begin{align*} g_{small}^{-1}Rf_{small, *}K & = i_g^{-1}\pi_S^{-1}Rf_{small, *}K \\ & \to i_g^{-1}Rf_{big, *}\pi_T^{-1}K \\ & = Rf'_{small, *}(i_{g'}^{-1}\pi_T^{-1}K) \\ & = Rf'_{small, *}((g'_{small})^{-1}K) \end{align*} where we have used $\pi_S \circ i_g = g_{small}$, $\pi_T \circ i_{g'} = g'_{small}$, and Lemma \ref{lemma-compare-higher-direct-image}. This map is an isomorphism by the proper base change theorem (Lemma \ref{lemma-proper-base-change}) provided $K$ is bounded below and the cohomology sheaves of $K$ are torsion. \medskip\noindent The proof of part (3) is the same as the proof of part (2), except we use Lemma \ref{lemma-proper-base-change-mod-n} instead of Lemma \ref{lemma-proper-base-change}. \medskip\noindent Proof of (4). If $f$ is finite, then the functors $f_{small, *}$ and $f_{big, *}$ are exact. This follows from Proposition \ref{proposition-finite-higher-direct-image-zero} for $f_{small}$. Since any base change $f'$ of $f$ is finite too, we conclude from Lemma \ref{lemma-compare-higher-direct-image} part (3) that $f_{big, *}$ is exact too (as the higher derived functors are zero). Thus this case follows from part (1).
You can prove this by repeating the proof of Lemma \ref{lemma-compare-higher-direct-image-proper} part (1); we will instead deduce the result from this. As $\epsilon_{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. The description in Lemma \ref{lemma-describe-pullback-pi-fppf} shows that $\epsilon_{Y, *} \circ a_Y^{-1} = \pi_Y^{-1}$ and similarly for $X$. To show that the canonical map $a_Y^{-1}f_{small, *}\mathcal{F} \to f_{big, fppf, *}a_X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that \begin{align*} \pi_Y^{-1}f_{small, *}\mathcal{F} & = \epsilon_{Y, *}a_Y^{-1}f_{small, *}\mathcal{F} \\ & \to  \epsilon_{Y, *}f_{big, fppf, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *} \epsilon_{X, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *}\pi_X^{-1}\mathcal{F} \end{align*} is an isomorphism. This is part (1) of Lemma \ref{lemma-compare-higher-direct-image-proper}.
We first show that $\mathcal{A}'_X \subset \textit{Ab}((\Sch/X)_\etale)$ is a weak Serre subcategory by checking conditions (1), (2), (3), and (4) of Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}. Parts (1), (2), (3) are immediate as $\pi_X^{-1}$ is exact and fully faithful for example by Lemma \ref{lemma-cohomological-descent-etale}. If $0 \to \pi_X^{-1}\mathcal{F} \to \mathcal{G} \to \pi_X^{-1}\mathcal{F}' \to 0$ is a short exact sequence in $\textit{Ab}((\Sch/X)_\etale)$ then $0 \to \mathcal{F} \to \pi_{X, *}\mathcal{G} \to \mathcal{F}' \to 0$ is exact by Lemma \ref{lemma-cohomological-descent-etale}. Hence $\mathcal{G} = \pi_X^{-1}\pi_{X, *}\mathcal{G}$ is in $\mathcal{A}'_X$ which checks the final condition. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-base-change-P}) holds by the existence of fibre products of schemes and the fact that the base change of a finite morphism of schemes is a finite morphism of schemes, see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-restriction-A}) follows from the commutative diagram (3) in Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-A-sheaf}) is Lemma \ref{lemma-describe-pullback-pi-fppf}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-A-and-P}) holds by Lemma \ref{lemma-compare-higher-direct-image-proper} part (4). \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-refine-tau-by-P}) is implied by More on Morphisms, Lemma \ref{more-morphisms-lemma-dominate-fppf-etale-locally}.
By Lemma \ref{lemma-compare-fppf-etale} the lemmas in Cohomology on Sites, Section \ref{sites-cohomology-section-compare-general} all apply to our current setting. To translate the results observe that the category $\mathcal{A}_X$ of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-A} is the essential image of $a_X^{-1} : \textit{Ab}(X_\etale) \to \textit{Ab}((\Sch/X)_{fppf})$. \medskip\noindent Part (1) is equivalent to $(V_n)$ for all $n$ which holds by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-C-all-n-general}. \medskip\noindent Part (2) follows by applying $\epsilon_Y^{-1}$ to the conclusion of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-C-general}. \medskip\noindent Part (3) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-C-all-n-general} part (1) because $\pi_X^{-1}K$ is in $D^+_{\mathcal{A}'_X}((\Sch/X)_\etale)$ and $a_X^{-1} = \epsilon_X^{-1} \circ a_X^{-1}$. \medskip\noindent Part (4) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-C-all-n-general} part (2) for the same reason. \medskip\noindent Part (5). We use that \begin{align*} R\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K & = Rf_{big, \etale, *}R\epsilon_{X, *}a_X^{-1}K \\ & = Rf_{big, \etale, *}\pi_X^{-1}K \\ & = \pi_Y^{-1}Rf_{small, *}K \\ & = R\epsilon_{Y, *} a_Y^{-1}Rf_{small, *}K \end{align*} The first equality by the commutative diagram in Lemma \ref{lemma-push-pull-fppf-etale} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pushforward-composition}. The second equality is (3). The third is Lemma \ref{lemma-compare-higher-direct-image-proper} part (2). The fourth is (3) again. Thus the base change map $a_Y^{-1}(Rf_{small, *}K) \to Rf_{big, fppf, *}(a_X^{-1}K)$ induces an isomorphism $$ R\epsilon_{Y, *}a_Y^{-1}Rf_{small, *}K \to R\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K $$ The proof is finished by the following remark: a map $\alpha : a_Y^{-1}L \to M$ with $L$ in $D^+(Y_\etale)$ and $M$ in $D^+((\Sch/Y)_{fppf})$ such that $R\epsilon_{Y, *}\alpha$ is an isomorphism, is an isomorphism. Namely,  we show by induction on $i$ that $H^i(\alpha)$ is an isomorphism. This is true for all sufficiently small $i$. If it holds for $i \leq i_0$, then we see that $R^j\epsilon_{Y, *}H^i(M) = 0$ for $j > 0$ and $i \leq i_0$ by (1) because $H^i(M) = a_Y^{-1}H^i(L)$ in this range. Hence $\epsilon_{Y, *}H^{i_0 + 1}(M) = H^{i_0 + 1}(R\epsilon_{Y, *}M)$ by a spectral sequence argument. Thus $\epsilon_{Y, *}H^{i_0 + 1}(M) = \pi_Y^{-1}H^{i_0 + 1}(L) = \epsilon_{Y, *}a_Y^{-1}H^{i_0 + 1}(L)$. This implies $H^{i_0 + 1}(\alpha)$ is an isomorphism (because $\epsilon_{Y, *}$ reflects isomorphisms as it is the identity on underlying presheaves) as desired.
We first reduce the statement to the case where $K$ is given by a single abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet$. By the case of a sheaf we see that $\mathcal{F}^n = a_{X, *} a_X^{-1} \mathcal{F}^n$ and that the sheaves $R^qa_{X, *}a_X^{-1}\mathcal{F}^n$ are zero for $q > 0$. By Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) applied to $a_X^{-1}\mathcal{F}^\bullet$ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$. \medskip\noindent By Lemma \ref{lemma-describe-pullback-pi-fppf} we have $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$. Thus it suffices to show that $R^qa_{X, *}a_X^{-1}\mathcal{F} = 0$ for $q > 0$. For this we can use $a_X = \epsilon_X \circ \pi_X$ and the Leray spectral sequence (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray}). By Lemma \ref{lemma-V-C-all-n-etale-fppf} we have $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$ and $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$. By Lemma \ref{lemma-cohomological-descent-etale} we have $R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof.
We urge the reader to find their own proof of these results based on the material in Descent, Section \ref{descent-section-quasi-coherent-sheaves}. \medskip\noindent We first explain why the notation in this lemma is consistent with our earlier use of the notation $\mathcal{F}^a$ in Sections \ref{section-quasi-coherent} and \ref{section-cohomology-quasi-coherent} and in Descent, Section \ref{descent-section-quasi-coherent-sheaves}. Namely, we know by Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} that there exists a quasi-coherent module $\mathcal{F}_0$ on the scheme $S$ (in other words on the small Zariski site) such that $\mathcal{F}$ is the restriction of the rule $$ \mathcal{F}_0^a : (\Sch/S)_\etale \longrightarrow \textit{Ab},\quad (f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F}) $$ to the subcategory $S_\etale \subset (\Sch/S)_\etale$ where here $f^*$ denotes usual pullback of sheaves of modules on schemes. Since $\mathcal{F}_0^a$ is pullback by the morphism of ringed sites $$ ((\Sch/S)_\etale, \mathcal{O}) \longrightarrow (S_{Zar}, \mathcal{O}_{S_{Zar}}) $$ by Descent, Remark \ref{descent-remark-change-topologies-ringed-sites} it follows immediately (from composition of pullbacks) that $\mathcal{F}^a = \mathcal{F}_0^a$. This proves the sheaf property even for fpqc coverings by Descent, Lemma \ref{descent-lemma-sheaf-condition-holds} (see also Proposition \ref{proposition-quasi-coherent-sheaf-fpqc}). Then (2) and (3) follow again by Descent, Remark \ref{descent-remark-change-topologies-ringed-sites} and (4) and (5) follow from Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} (see also the meta result Theorem \ref{theorem-quasi-coherent}). \medskip\noindent Part (6) is immediate from the description of the sheaf $\mathcal{F}^a = \pi_S^*\mathcal{F} = a_S^*\mathcal{F}$. \medskip\noindent For any abelian $\mathcal{H}$ on $(\Sch/S)_{fppf}$ the higher direct image $R^p\epsilon_{S, *}\mathcal{H}$ is the sheaf associated to the presheaf $U \mapsto H^p_{fppf}(U, \mathcal{H})$ on $(\Sch/S)_\etale$. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. Hence to prove $R^p\epsilon_{S, *}a_S^*\mathcal{F} = R^p\epsilon_{S, *}\mathcal{F}^a = 0$ for $p > 0$ it suffices to show that any scheme $U$ over $S$ has an \'etale covering $\{U_i \to U\}_{i \in I}$ such that $H^p_{fppf}(U_i, \mathcal{F}^a) = 0$ for $p > 0$. If we take an open covering by affines, then the required vanishing follows from comparison with usual cohomology (Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent} or Theorem \ref{theorem-zariski-fpqc-quasi-coherent}) and the vanishing of cohomology of quasi-coherent sheaves on affine schemes afforded by Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. \medskip\noindent To show that $R^pa_{S, *}a_S^{-1}\mathcal{F} = R^pa_{S, *}\mathcal{F}^a = 0$ for $p > 0$ we argue in exactly the same manner. This finishes the proof.
The statement about the \'etale topology is the content of Lemma \ref{lemma-describe-pullback}. To finish the proof it suffices to show that $\pi_S^{-1}\mathcal{F}$ is a sheaf for the ph topology. By Topologies, Lemma \ref{topologies-lemma-characterize-sheaf} it suffices to show that given a proper surjective morphism $V \to U$ of schemes over $S$ we have an equalizer diagram $$ \xymatrix{ (\pi_S^{-1}\mathcal{F})(U) \ar[r] & (\pi_S^{-1}\mathcal{F})(V) \ar@<1ex>[r] \ar@<-1ex>[r] & (\pi_S^{-1}\mathcal{F})(V \times_U V) } $$ Set $\mathcal{G} = \pi_S^{-1}\mathcal{F}|_{U_\etale}$. Consider the commutative diagram $$ \xymatrix{ V \times_U V \ar[r] \ar[rd]_g \ar[d] & V \ar[d]^f \\ V \ar[r]^f & U } $$ We have $$ (\pi_S^{-1}\mathcal{F})(V) = \Gamma(V, f^{-1}\mathcal{G}) = \Gamma(U, f_*f^{-1}\mathcal{G}) $$ where we use $f_*$ and $f^{-1}$ to denote functorialities between small \'etale sites. Second, we have $$ (\pi_S^{-1}\mathcal{F})(V \times_U V) = \Gamma(V \times_U V, g^{-1}\mathcal{G}) = \Gamma(U, g_*g^{-1}\mathcal{G}) $$ The two maps in the equalizer diagram come from the two maps $$ f_*f^{-1}\mathcal{G} \longrightarrow g_*g^{-1}\mathcal{G} $$ Thus it suffices to prove $\mathcal{G}$ is the equalizer of these two maps of sheaves. Let $\overline{u}$ be a geometric point of $U$. Set $\Omega = \mathcal{G}_{\overline{u}}$. Taking stalks at $\overline{u}$ by Lemma \ref{lemma-proper-pushforward-stalk} we obtain the two maps $$ H^0(V_{\overline{u}}, \underline{\Omega}) \longrightarrow H^0((V \times_U V)_{\overline{u}}, \underline{\Omega}) = H^0(V_{\overline{u}} \times_{\overline{u}} V_{\overline{u}}, \underline{\Omega}) $$ where $\underline{\Omega}$ indicates the constant sheaf with value $\Omega$. Of course these maps are the pullback by the projection maps. Then it is clear that the sections coming from pullback by projection onto the first factor are constant on the fibres of the first projection, and sections coming from pullback by projection onto the first factor are constant on the fibres of the first projection. The sections in the intersection of the images of these pullback maps are constant on all of $V_{\overline{u}} \times_{\overline{u}} V_{\overline{u}}$, i.e., these come from elements of $\Omega$ as desired.
You can prove this by repeating the proof of Lemma \ref{lemma-compare-higher-direct-image-proper} part (1); we will instead deduce the result from this. As $\epsilon_{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. The description in Lemma \ref{lemma-describe-pullback-pi-ph} shows that $\epsilon_{Y, *} \circ a_Y^{-1} = \pi_Y^{-1}$ and similarly for $X$. To show that the canonical map $a_Y^{-1}f_{small, *}\mathcal{F} \to f_{big, ph, *}a_X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that \begin{align*} \pi_Y^{-1}f_{small, *}\mathcal{F} & = \epsilon_{Y, *}a_Y^{-1}f_{small, *}\mathcal{F} \\ & \to  \epsilon_{Y, *}f_{big, ph, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *} \epsilon_{X, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *}\pi_X^{-1}\mathcal{F} \end{align*} is an isomorphism. This is part (1) of Lemma \ref{lemma-compare-higher-direct-image-proper}.
We first show that $\mathcal{A}'_X \subset \textit{Ab}((\Sch/X)_\etale)$ is a weak Serre subcategory by checking conditions (1), (2), (3), and (4) of Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}. Parts (1), (2), (3) are immediate as $\pi_X^{-1}$ is exact and fully faithful for example by Lemma \ref{lemma-cohomological-descent-etale}. If $0 \to \pi_X^{-1}\mathcal{F} \to \mathcal{G} \to \pi_X^{-1}\mathcal{F}' \to 0$ is a short exact sequence in $\textit{Ab}((\Sch/X)_\etale)$ then $0 \to \mathcal{F} \to \pi_{X, *}\mathcal{G} \to \mathcal{F}' \to 0$ is exact by Lemma \ref{lemma-cohomological-descent-etale}. In particular we see that $\pi_{X, *}\mathcal{G}$ is an abelian torsion sheaf on $X_\etale$. Hence $\mathcal{G} = \pi_X^{-1}\pi_{X, *}\mathcal{G}$ is in $\mathcal{A}'_X$ which checks the final condition. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-base-change-P}) holds by the existence of fibre products of schemes and the fact that the base change of a proper morphism of schemes is a proper morphism of schemes, see Morphisms, Lemma \ref{morphisms-lemma-base-change-proper}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-restriction-A}) follows from the commutative diagram (3) in Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-A-sheaf}) is Lemma \ref{lemma-describe-pullback-pi-ph}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-A-and-P}) holds by Lemma \ref{lemma-compare-higher-direct-image-proper} part (2) and the fact that $R^if_{small}\mathcal{F}$ is torsion if $\mathcal{F}$ is an abelian torsion sheaf on $X_\etale$, see Lemma \ref{lemma-torsion-direct-image}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-refine-tau-by-P}) follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-dominate-fppf-etale-locally} combined with the fact that a finite morphism is proper and a surjective proper morphism defines a ph covering, see Topologies, Lemma \ref{topologies-lemma-surjective-proper-ph}.
By Lemma \ref{lemma-compare-ph-etale} the lemmas in Cohomology on Sites, Section \ref{sites-cohomology-section-compare-general} all apply to our current setting. To translate the results observe that the category $\mathcal{A}_X$ of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-A} is the full subcategory of $\textit{Ab}((\Sch/X)_{ph})$ consisting of sheaves of the form $a_X^{-1}\mathcal{F}$ where $\mathcal{F}$ is an abelian torsion sheaf on $X_\etale$. \medskip\noindent Part (1) is equivalent to $(V_n)$ for all $n$ which holds by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-C-all-n-general}. \medskip\noindent Part (2) follows by applying $\epsilon_Y^{-1}$ to the conclusion of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-C-general}. \medskip\noindent Part (3) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-C-all-n-general} part (1) because $\pi_X^{-1}K$ is in $D^+_{\mathcal{A}'_X}((\Sch/X)_\etale)$ and $a_X^{-1} = \epsilon_X^{-1} \circ a_X^{-1}$. \medskip\noindent Part (4) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-C-all-n-general} part (2) for the same reason.
We first reduce the statement to the case where $K$ is given by a single abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet$ of torsion abelian sheaves. This is possible by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-torsion}. By the case of a sheaf we see that $\mathcal{F}^n = a_{X, *} a_X^{-1} \mathcal{F}^n$ and that the sheaves $R^qa_{X, *}a_X^{-1}\mathcal{F}^n$ are zero for $q > 0$. By Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) applied to $a_X^{-1}\mathcal{F}^\bullet$ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$ where $\mathcal{F}$ is a torsion abelian sheaf. \medskip\noindent By Lemma \ref{lemma-describe-pullback-pi-ph} we have $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$. Thus it suffices to show that $R^qa_{X, *}a_X^{-1}\mathcal{F} = 0$ for $q > 0$. For this we can use $a_X = \epsilon_X \circ \pi_X$ and the Leray spectral sequence (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray}). By Lemma \ref{lemma-V-C-all-n-etale-ph} we have $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$ and $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$. By Lemma \ref{lemma-cohomological-descent-etale} we have $R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof.
The statement about the \'etale topology is the content of Lemma \ref{lemma-describe-pullback}. To finish the proof it suffices to show that $\pi_S^{-1}\mathcal{F}$ is a sheaf for the h topology. However, in Lemma \ref{lemma-describe-pullback-pi-ph} we have shown that $\pi_S^{-1}\mathcal{F}$ is a sheaf even in the stronger ph topology.
You can prove this by repeating the proof of Lemma \ref{lemma-compare-higher-direct-image-proper} part (1); we will instead deduce the result from this. As $\epsilon_{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. The description in Lemma \ref{lemma-describe-pullback-pi-h} shows that $\epsilon_{Y, *} \circ a_Y^{-1} = \pi_Y^{-1}$ and similarly for $X$. To show that the canonical map $a_Y^{-1}f_{small, *}\mathcal{F} \to f_{big, h, *}a_X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that \begin{align*} \pi_Y^{-1}f_{small, *}\mathcal{F} & = \epsilon_{Y, *}a_Y^{-1}f_{small, *}\mathcal{F} \\ & \to  \epsilon_{Y, *}f_{big, h, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *} \epsilon_{X, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *}\pi_X^{-1}\mathcal{F} \end{align*} is an isomorphism. This is part (1) of Lemma \ref{lemma-compare-higher-direct-image-proper}.
We first show that $\mathcal{A}'_X \subset \textit{Ab}((\Sch/X)_\etale)$ is a weak Serre subcategory by checking conditions (1), (2), (3), and (4) of Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}. Parts (1), (2), (3) are immediate as $\pi_X^{-1}$ is exact and fully faithful for example by Lemma \ref{lemma-cohomological-descent-etale}. If $0 \to \pi_X^{-1}\mathcal{F} \to \mathcal{G} \to \pi_X^{-1}\mathcal{F}' \to 0$ is a short exact sequence in $\textit{Ab}((\Sch/X)_\etale)$ then $0 \to \mathcal{F} \to \pi_{X, *}\mathcal{G} \to \mathcal{F}' \to 0$ is exact by Lemma \ref{lemma-cohomological-descent-etale}. In particular we see that $\pi_{X, *}\mathcal{G}$ is an abelian torsion sheaf on $X_\etale$. Hence $\mathcal{G} = \pi_X^{-1}\pi_{X, *}\mathcal{G}$ is in $\mathcal{A}'_X$ which checks the final condition. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-base-change-P}) holds by the existence of fibre products of schemes, the fact that the base change of a proper morphism of schemes is a proper morphism of schemes, see Morphisms, Lemma \ref{morphisms-lemma-base-change-proper}, and the fact that the base change of a morphism of finite presentation is a morphism of finite presentation, see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-presentation}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-restriction-A}) follows from the commutative diagram (3) in Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-A-sheaf}) is Lemma \ref{lemma-describe-pullback-pi-h}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-A-and-P}) holds by Lemma \ref{lemma-compare-higher-direct-image-proper} part (2) and the fact that $R^if_{small}\mathcal{F}$ is torsion if $\mathcal{F}$ is an abelian torsion sheaf on $X_\etale$, see Lemma \ref{lemma-torsion-direct-image}. \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-refine-tau-by-P}) is implied by More on Morphisms, Lemma \ref{more-morphisms-lemma-dominate-fppf-etale-locally} combined with the fact that a surjective finite locally free morphism is surjective, proper, and of finite presentation and hence defines a h-covering by More on Flatness, Lemma \ref{flat-lemma-surjective-proper-finite-presentation-h}.