Lemma 21.20.11. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be a map of sheaves of rings. If $\mathcal{I}^\bullet $ is a K-injective complex of $\mathcal{O}$-modules, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}', \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}'$-modules.

**Proof.**
This is true because $\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}')}(\mathcal{G}^\bullet , \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{O}', \mathcal{I}^\bullet )) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O})}(\mathcal{G}^\bullet , \mathcal{I}^\bullet )$ by Modules on Sites, Lemma 18.27.8.
$\square$

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## Comments (2)

Comment #2117 by Kestutis Cesnavicius on

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