The Stacks project

Lemma 61.26.4. Let $j : U \to X$ be an open immersion of schemes. Then $\text{id} \cong j^{-1}j_!$ and $j^{-1}j_* \cong \text{id}$ and the functors $j_!$ and $j_*$ are fully faithful.

Proof. See Modules on Sites, Lemma 18.19.8 (and Sites, Lemma 7.27.4 for the case of sheaves of sets) and Categories, Lemma 4.24.4. $\square$

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