Relative functors from categories with cofibrations

Example 20.1.9. More generally, let \((C,W,I)\) be an \(\infty \)-category with weak equivalences and cofibrations, in the sense of Definition 2.2.4, and consider a functor \(F\colon C \to D\) for which the restriction \(F\vert _{C_c}\colon C_c \to D\) to the cofibrant objects inverts weak equivalences. It follows from Proposition 6.6.3 that \(F\) admits an absolute left derived functor \(\bL F\colon C[W^{-1}] \to D\), uniquely characterized by the existence of an isomorphism \(\bL F\circ \gamma \circ i\simeq F\circ i\) on the cofibrant objects.