Lemma 13.17.6. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined.

The functors $R^ iF$, $i \geq 0$ come equipped with a canonical structure of a $\delta $-functor from $\mathcal{A} \to \mathcal{B}$, see Homology, Definition 12.11.1.

If every object of $\mathcal{A}$ is a subobject of a right acyclic object for $F$, then $\{ R^ iF, \delta \} _{i \geq 0}$ is a universal $\delta $-functor, see Homology, Definition 12.11.3.

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