Remark 6.6.4. Everything in the above proof is formal except for the statement being cited: that the functor \(\bL F\), which by construction only knows about the cofibrant objects, is a right Kan extension of \(F\) along \(\gamma \). The latter is a statement about all objects of \(C\), and it is here that the real work happens: one needs to know that Kan extensions along \(\gamma \) may be computed in terms of the cofibrant objects. This is [Cisinski (2019), Corollary 7.5.17], whose proof rests on the theory of finite direct diagrams and their Reedy fibrant replacements developed in [Cisinski (2019), Section 7.4].
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