This is similar to Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale} (3) but there is a small snag due to the fact that $F \to G$ may not be representable by schemes. In particular we don't get a commutative diagram of ringed sites, but only a commutative diagram of ringed topoi. \medskip\noindent Before we start the proof proper, we choose equivalences $j : \mathcal{S}_F \to \mathcal{X}$ and $j' : \mathcal{S}_G \to \mathcal{Y}$ which induce functors $u : F_\etale \to \mathcal{X}$ and $u' : G_\etale \to \mathcal{Y}$ as in the proof of Lemma \ref{lemma-compare}. Because of the 2-functoriality of sheaves on categories fibred in groupoids over $\Sch_{fppf}$ (see discussion in Section \ref{section-presheaves}) we may assume that $\mathcal{X} = \mathcal{S}_F$ and $\mathcal{Y} = \mathcal{S}_G$ and that $f : \mathcal{S}_F \to \mathcal{S}_G$ is the functor associated to the morphism $f : F \to G$. Correspondingly we will omit $u$ and $u'$ from the notation, i.e., given an object $U \to F$ of $F_\etale$ we denote $U/F$ the corresponding object of $\mathcal{X}$. Similarly for $G$. \medskip\noindent Let $\mathcal{G}$ be a sheaf on $\mathcal{X}_\etale$. To prove (2) we compute $\pi_{G, *}f_*\mathcal{G}$ and $f_{small, *}\pi_{F, *}\mathcal{G}$. To do this let $V \to G$ be an object of $G_\etale$. Then $$ \pi_{G, *}f_*\mathcal{G}(V) = f_*\mathcal{G}(V/G) = \Gamma\Big( (\Sch/V)_{fppf} \times_{\mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{G}\Big) $$ see (\ref{equation-pushforward}). The fibre product in the formula is $$ (\Sch/V)_{fppf} \times_{\mathcal{Y}} \mathcal{X} = (\Sch/V)_{fppf} \times_{\mathcal{S}_G} \mathcal{S}_F = \mathcal{S}_{V \times_G F} $$ i.e., it is the split category fibred in groupoids associated to the algebraic space $V \times_G F$. And $\text{pr}^{-1}\mathcal{G}$ is a sheaf on $\mathcal{S}_{V \times_G F}$ for the \'etale topology. \medskip\noindent In particular, if $V \times_G F$ is representable, i.e., if it is a scheme, then $\pi_{G, *}f_*\mathcal{G}(V) = \mathcal{G}(V \times_G F/F)$ and also $$ f_{small, *}\pi_{F, *}\mathcal{G}(V) = \pi_{F, *}\mathcal{G}(V \times_G F) = \mathcal{G}(V \times_G F/F) $$ which proves the desired equality in this special case. \medskip\noindent In general, choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_G F$. Set $R = U \times_{V \times_G F} U$. Then $U/V \times_G F$ and $R/V \times_G F$ are objects of the fibre product category above. Since $\text{pr}^{-1}\mathcal{G}$ is a sheaf for the \'etale topology on $\mathcal{S}_{V \times_G F}$ the diagram $$ \xymatrix{ \Gamma\Big( (\Sch/V)_{fppf} \times_{\mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{G}\Big) \ar[r] & \text{pr}^{-1}\mathcal{G}(U/V \times_G F) \ar@<1ex>[r] \ar@<-1ex>[r] & \text{pr}^{-1}\mathcal{G}(R/V \times_G F) } $$ is an equalizer diagram. Note that $\text{pr}^{-1}\mathcal{G}(U/V \times_G F) = \mathcal{G}(U/F)$ and $\text{pr}^{-1}\mathcal{G}(R/V \times_G F) = \mathcal{G}(R/F)$ by the definition of pullbacks. Moreover, by the material in Properties of Spaces, Section \ref{spaces-properties-section-etale-site} (especially, Properties of Spaces, Remark \ref{spaces-properties-remark-explain-equivalence} and Lemma \ref{spaces-properties-lemma-functoriality-etale-site}) we see that there is an equalizer diagram $$ \xymatrix{ f_{small, *}\pi_{F, *}\mathcal{G}(V) \ar[r] & \pi_{F, *}\mathcal{G}(U/F) \ar@<1ex>[r] \ar@<-1ex>[r] & \pi_{F, *}\mathcal{G}(R/F) } $$ Since we also have $\pi_{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$ and $\pi_{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$ we obtain a canonical identification $f_{small, *}\pi_{F, *}\mathcal{G}(V) = \pi_{G, *}f_*\mathcal{G}(V)$. We omit the proof that this is compatible with restriction mappings and that it is functorial in $\mathcal{G}$.