The Stacks project

Lemma 61.26.2. Let $j : U \to X$ be an étale morphism of schemes. Let $\mathcal{G}$ be an abelian sheaf on $U_{\acute{e}tale}$. Then $\epsilon ^{-1} j_!\mathcal{G} = j_!\epsilon ^{-1}\mathcal{G}$ as sheaves on $X_{pro\text{-}\acute{e}tale}$.

Proof. This is true because both are left adjoints to $j_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1} = \epsilon ^{-1}j_{{\acute{e}tale}, *}$, see Lemma 61.23.1. $\square$

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