Lemma 34.4.12. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$. The functor $S_{affine, {\acute{e}tale}} \to S_{\acute{e}tale}$ is special cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits (S_{affine, {\acute{e}tale}})$ to $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale})$.

Proof. Omitted. Hint: compare with the proof of Lemma 34.4.11. $\square$

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