Lemma 63.3.15. Let $f : X \to Y$ be morphism of schemes which is separated and locally of finite type. Let $X = \bigcup _{i \in I} X_ i$ be an open covering such that for all $i, j \in I$ there exists a $k$ with $X_ i \cup X_ j \subset X_ k$. Denote $f_ i : X_ i \to Y$ the restriction of $f$. Then

\[ f_!\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} f_{i, !}(\mathcal{F}|_{X_ i}) \]

functorially in $\mathcal{F} \in \textit{Ab}(X_{\acute{e}tale})$ where the transition maps are the ones constructed in Remark 63.3.5.

**Proof.**
It suffices to show that the canonical map from right to left is a bijection when evaluated on a quasi-compact object $V$ of $Y_{\acute{e}tale}$. Observe that the colimit on the right hand side is directed and has injective transition maps. Thus we can use Sites, Lemma 7.17.7 to evaluate the colimit. Hence, the statement comes down to the observation that a closed subset $Z \subset X_ V$ proper over $V$ is quasi-compact and hence is contained in $X_{i, V}$ for some $i$.
$\square$

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