Lemma 34.4.13. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{\acute{e}tale}$. The functor $T_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$ is cocontinuous and induces a morphism of topoi
\[ i_ f : \mathop{\mathit{Sh}}\nolimits (T_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale}) \]
For a sheaf $\mathcal{G}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$ we have the formula $(i_ f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$. The functor $i_ f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.
Proof.
Denote the functor $u : T_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$. In other words, given an étale morphism $j : U \to T$ corresponding to an object of $T_{\acute{e}tale}$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma 34.4.10. Let $a, b : U \to V$ be two morphisms in $T_{\acute{e}tale}$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is
\[ V \times _{\Delta _{V/T}, V \times _ T V, (a, b)} U \]
which is a fibre product of schemes étale over $T$, hence étale over $T$. Thus $T_{\acute{e}tale}$ 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 7.21.5 and 7.21.6.
$\square$
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