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.
The functor RF is an exact functor D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}).
The functor RF induces an exact functor K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}).
The functor RF induces a \delta -functor \text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}).
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:
The derived functor is exact: This boils down to Lemma 13.14.6.
This is true because K^{+}(\mathcal{A}) \to D^{+}(\mathcal{A}) is exact and compositions of exact functors are exact.
This is true because \text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{A}) is a \delta -functor, see Lemma 13.12.1.
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.
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