The Stacks project

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

1. 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.

2. 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.