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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