TITLE: Simplicial localization of the cofibrant-fibrant objects
QUESTION [6 upvotes]: Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant objects (respectively, the cofibrant and fibrant objects).
Does the inclusion $M^{cf} \to M^c$ induce a Dwyer-Kan equivalence on the simplicial localizations at the class of weak equivalences?
Assuming the answer is "yes", does anyone know of a reference? I couldn't find this in the original papers of Dwyer and Kan for general $M$, only when $M$ has functorial factorizations.

REPLY [2 votes]: The only published reference for the full statement I know is Hinich's paper (V. Hinich, Dwyer-Kan localization revisited. Homology Homotopy Appl. 18, No. 1), where it appears as (the dual of) Proposition 1.3.8. Strictly speaking, the proof given there is "this works just like the proof of the case $\mathscr C^c\hookrightarrow\mathscr C$ (of which Hinich gives a new proof in Section 1.3.7), but at least to me it looks like you can indeed directly adapt his argument by just adding the word "fibrant" at appropriate places.