Lemma 13.20.3. Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor.

1. The functor $RF$ is an exact functor $D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.

2. The functor $RF$ induces an exact functor $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.

3. The functor $RF$ induces a $\delta$-functor $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.

4. The functor $RF$ induces a $\delta$-functor $\mathcal{A} \to D^{+}(\mathcal{B})$.

Proof. This lemma simply reviews some of the results obtained so far. Note that by Lemma 13.20.2 $RF$ is everywhere defined. Here are some references:

1. The derived functor is exact: This boils down to Lemma 13.14.6.

2. This is true because $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ is exact and compositions of exact functors are exact.

3. This is true because $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ is a $\delta$-functor, see Lemma 13.12.1.

4. This is true because $\mathcal{A} \to \text{Comp}^{+}(\mathcal{A})$ is exact and precomposing a $\delta$-functor by an exact functor gives a $\delta$-functor.

$\square$

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