Lemma 24.29.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Then any exact functor
\[ T : K(\textit{Mod}(\mathcal{A}, \text{d})) \longrightarrow \mathcal{D} \]
of triangulated categories has a right derived extension $RT : D(\mathcal{A}, \text{d}) \to \mathcal{D}$ whose value on a graded injective and K-injective differential graded $\mathcal{A}$-module $\mathcal{I}$ is $T(\mathcal{I})$.
Proof.
By Theorem 24.25.13 for any (right) differential graded $\mathcal{A}$-module $\mathcal{M}$ there exists a quasi-isomorphism $\mathcal{M} \to \mathcal{I}$ where $\mathcal{I}$ is a graded injective and K-injective differential graded $\mathcal{A}$-module. Hence by Derived Categories, Lemma 13.14.15 it suffices to show that given a quasi-isomorphism $\mathcal{I} \to \mathcal{I}'$ of differential graded $\mathcal{A}$-modules which are both graded injective and K-injective then $T(\mathcal{I}) \to T(\mathcal{I}')$ is an isomorphism. This is true because the map $\mathcal{I} \to \mathcal{I}'$ is an isomorphism in $K(\textit{Mod}(\mathcal{A}, \text{d}))$ as follows for example from Lemma 24.26.7 (or one can deduce it from Lemma 24.25.10).
$\square$
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