The Stacks project

Lemma 75.10.6. Let $S$ be a scheme. Let $j : U \to X$ be a ├ętale morphism of algebraic spaces over $S$. Given an ├ętale morphism $V \to Y$, set $W = V \times _ X U$ and denote $j_ W : W \to V$ the projection morphism. Then $(j_!E)|_ V = j_{W!}(E|_ W)$ for $E$ in $D(\mathcal{O}_ U)$.

Proof. This is true because $(j_!\mathcal{F})|_ V = j_{W!}(\mathcal{F}|_ W)$ for an $\mathcal{O}_ X$-module $\mathcal{F}$ as follows immediately from the construction of the functors $j_!$ and $j_{W!}$, see Modules on Sites, Lemma 18.19.2. $\square$

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