Lemma 85.3.6. In Situation 85.3.3. If $\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{C}_{total})$, then $\mathcal{I}_ n$ is injective in $\textit{Ab}(\mathcal{C}_ n)$. If $\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}(\mathcal{C}_{total})$, then $\mathcal{I}_ n^\bullet $ is K-injective in $\textit{Ab}(\mathcal{C}_ n)$.

** An injective abelian sheaf on a simplicial site is injective on each component **

**Proof.**
The first statement follows from Homology, Lemma 12.29.1 and Lemma 85.3.5. The second statement from Derived Categories, Lemma 13.31.9 and Lemma 85.3.5.
$\square$

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