The Stacks project

Lemma 21.10.1. Let $\mathcal{C}$ be a site. An injective abelian sheaf is also injective as an object in the category $\textit{PAb}(\mathcal{C})$.

Proof. Apply Homology, Lemma 12.29.1 to the categories $\mathcal{A} = \textit{Ab}(\mathcal{C})$, $\mathcal{B} = \textit{PAb}(\mathcal{C})$, the inclusion functor and sheafification. (See Modules on Sites, Section 18.3 to see that all assumptions of the lemma are satisfied.) $\square$

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