Lemma 61.31.2. Let $S$ be a scheme. Let $S_{pro\text{-}\acute{e}tale}\subset S_{pro\text{-}\acute{e}tale}'$ be two small pro-étale sites of $S$ as constructed in Definition 61.12.8. Then the inclusion functor satisfies the assumptions of Sites, Lemma 7.21.8. Hence there exist morphisms of topoi

whose composition is isomorphic to the identity and with $f_* = g^{-1}$. Moreover,

for $\mathcal{F}' \in \textit{Ab}(S_{pro\text{-}\acute{e}tale}')$ we have $H^ p(S_{pro\text{-}\acute{e}tale}', \mathcal{F}') = H^ p(S_{pro\text{-}\acute{e}tale}, g^{-1}\mathcal{F}')$,

for $\mathcal{F} \in \textit{Ab}(S_{pro\text{-}\acute{e}tale})$ we have

\[ H^ p(S_{pro\text{-}\acute{e}tale}, \mathcal{F}) = H^ p(S_{pro\text{-}\acute{e}tale}', g_*\mathcal{F}) = H^ p(S_{pro\text{-}\acute{e}tale}', f^{-1}\mathcal{F}). \]

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