Lemma 13.20.2. Let $\mathcal{A}$ be an abelian category with enough injectives.

For any exact functor $F : K^{+}(\mathcal{A}) \to \mathcal{D}$ into a triangulated category $\mathcal{D}$ the right derived functor

\[ RF : D^{+}(\mathcal{A}) \longrightarrow \mathcal{D} \]is everywhere defined.

For any additive functor $F : \mathcal{A} \to \mathcal{B}$ into an abelian category $\mathcal{B}$ the right derived functor

\[ RF : D^{+}(\mathcal{A}) \longrightarrow D^{+}(\mathcal{B}) \]is everywhere defined.

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