Lemma 63.4.5. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Then

for $\mathcal{F}$ in $\textit{Ab}(X_{\acute{e}tale})$ and a geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ we have

\[ (f_!\mathcal{F})_{\overline{y}} = \bigoplus \nolimits _{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} \]functorially in $\mathcal{F}$, and

the functor $f_! : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is exact and commutes with direct sums.

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