Lemma 61.12.13. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. The inclusion functor $S_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ satisfies the hypotheses of Sites, Lemma 7.21.8 and hence induces a morphism of sites

\[ \pi _ S : (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\longrightarrow S_{pro\text{-}\acute{e}tale} \]

and a morphism of topoi

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

such that $\pi _ S \circ i_ S = \text{id}$. Moreover, $i_ S = i_{\text{id}_ S}$ with $i_{\text{id}_ S}$ as in Lemma 61.12.12. In particular the functor $i_ S^{-1} = \pi _{S, *}$ is described by the rule $i_ S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.

**Proof.**
In this case the functor $u : S_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, in addition to the properties seen in the proof of Lemma 61.12.12 above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 7.21.8.
$\square$

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