Definition 63.6.4. Let $F: \mathcal{A} \to \mathcal{B}$ be a left exact functor and assume that $\mathcal{A}$ has enough injectives. We define the *total right derived functor of $F$* as the functor $RF: D^+(\mathcal{A}) \to D^+(\mathcal{B})$ fitting into the diagram

This is possible since the left vertical arrow is invertible by the previous lemma. Similarly, let $G: \mathcal{A} \to \mathcal{B}$ be a right exact functor and assume that $\mathcal{A}$ has enough projectives. We define the *total left derived functor of $G$* as the functor $LG: D^-(\mathcal{A}) \to D^-(\mathcal{B})$ fitting into the diagram

This is possible since the left vertical arrow is invertible by the previous lemma.

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