The Stacks project

Lemma 21.12.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. An injective sheaf of modules is also injective as an object in the category $\textit{PMod}(\mathcal{O})$.

Proof. Apply Homology, Lemma 12.29.1 to the categories $\mathcal{A} = \textit{Mod}(\mathcal{O})$, $\mathcal{B} = \textit{PMod}(\mathcal{O})$, the inclusion functor and sheafification. (See Modules on Sites, Section 18.11 to see that all assumptions of the lemma are satisfied.) $\square$

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