Lemma 61.12.12. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. The functor $T_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is cocontinuous and induces a morphism of topoi

\[ i_ f : \mathop{\mathit{Sh}}\nolimits (T_{pro\text{-}\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}) \]

For a sheaf $\mathcal{G}$ on $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ 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 $u : T_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. In other words, given a weakly étale morphism $j : U \to T$ corresponding to an object of $T_{pro\text{-}\acute{e}tale}$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma 61.12.10. Moreover, $T_{pro\text{-}\acute{e}tale}$ has equalizers and $u$ commutes with them by Lemma 61.12.10. 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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