Remark 6.6.5. The situation simplifies considerably when \(C\) admits a functorial cofibrant replacement: a functor \(Q\colon C \to C_c\) together with a natural weak equivalence \(q\colon i \circ Q \Rightarrow \id _C\). Such a replacement exists as soon as \((C,W,I)\) admits functorial factorizations, by factoring the map \(\emptyset \to X\) functorially in \(X\). One may then simply take \(\bL F \circ \gamma \simeq F \circ Q\), with structure map \(F(q)\colon F \circ Q \Rightarrow F\), and the universal property can be deduced directly from the naturality of \(q\), without any of the machinery above. This is the route taken in [Dwyer et al. (2004), Sections 39-41] and [Riehl (2014), Chapter 2], where such data is called a left deformation.
We have avoided this hypothesis because it is genuinely restrictive: while \(\Top \) and \(\Ch (\Aa )\) do admit functorial factorizations, the projective cofibration structure on bounded below chain complexes from Proposition 6.6.15 does not, at least not for an arbitrary abelian category \(\Aa \) with enough projectives: a projective resolution is built from a choice of epimorphism \(P \twoheadrightarrow A\) from a projective object for each \(A \in \Aa \), and such choices cannot in general be made functorially. (For \(\Aa = \Mod _R\) one can of course take the free module on the underlying set of \(A\).)
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