Lemma 61.19.8. Let $X$ be a scheme. Let $\Lambda $ be a ring.

The essential image of the fully faithful functor $\epsilon ^{-1} : \textit{Mod}(X_{\acute{e}tale}, \Lambda ) \to \textit{Mod}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ is a weak Serre subcategory $\mathcal{C}$.

The functor $\epsilon ^{-1}$ defines an equivalence of categories of $D^+(X_{\acute{e}tale}, \Lambda )$ with $D^+_\mathcal {C}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ with question inverse given by $R\epsilon _*$.

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