Construction 6.6.17 (Derived functors between derived categories). Let \(\Aa \) and \(\Bb \) be abelian categories, assume that \(\Aa \) has enough projectives, and let \(F\colon \Aa \to \Bb \) be an additive functor. Extending \(F\) degreewise gives a functor \(\Ch ^-(F)\colon \Ch ^-(\Aa ) \to \Ch ^-(\Bb )\) that preserves chain homotopies. By Corollary 6.6.12, the restriction \[ \Ch ^-(F)\vert _{\proj }\colon \Ch ^-_{\proj }(\Aa ) \to \Ch ^-(\Bb ) \] preserves quasi-isomorphisms. Composing with the localization \(\gamma _{\Bb }\colon \Ch ^-(\Bb ) \to \D ^-(\Bb )\), we obtain a functor that inverts quasi-isomorphisms between the cofibrant objects of the projective cofibration structure on \(\Ch ^-(\Aa )\). By Proposition 6.6.3, it therefore admits a total left derived functor \[ \bL F \colon \D ^-(\Aa ) \to \D ^-(\Bb ) \] If \(q\colon P_{\bullet }\xrightarrow {\sim }C_{\bullet }\) is any projective resolution, then \[ \bL F(C_{\bullet })\simeq F(P_{\bullet }). \]
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