Definition 63.8.1. Let $T: \mathcal{A} \to \mathcal{B}$ be a left exact functor and assume that $\mathcal{A}$ has enough injectives. Define $RT: DF^+(\mathcal{A}) \to D F^+(\mathcal{B})$ to fit in the diagram

This is well-defined by the previous lemma. Let $G: \mathcal{A} \to \mathcal{B}$ be a right exact functor and assume that $\mathcal{A}$ has enough projectives. Define $LG: DF^+(\mathcal{A}) \to DF^+(\mathcal{B})$ to fit in the diagram

Again, this is well-defined by the previous lemma. The functors $RT$, resp. $LG$, are called the *filtered derived functor* of $T$, resp. $G$.

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