The Stacks project

Definition 64.8.1. 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. $G$.