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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