Infinity category of functors from a relative category to a model category

Question

Let $M$ be a model category (maybe cofibrantly generated/combinatorial). Let $(C,W)$ be a relative category.

I write $M^{(C,W)}$ for the full subcategory of $M^C$ on relative functors. This is a relative category if I take as weak equivalences the levelwise weak equivalences.

I am looking for a reference for the following fact:

$M^{(C,W)}$ is a model for the $\infty$-category of functors from $(C,W)$ to $M$.

The reason this is not trivial is that the nerve of $M$ in Rezk's sense is not quite a complete Segal space (in fact it was proved by Barwick and Kan that it is one up to a Reedy fibrant replacement). I know I have read this result somewhere in the literature but I can't remember where.

Answer 1

The following is taken from Section 2.3.2 of

http://arxiv.org/abs/math/0207028

Let $C$ be a small simplicial category, $S\subseteq C$ a simplicial subcategory and $M$ a simplicial combinatorial model category. Consider the category $M^C$ of simplicial functors from $C$ to $M$, endowed with the projective model structure.

Using $S$ and the set of generating cofibrations in $M$, we can define a specific set of morphisms in $M^C$ and Bousfield localize $M^C$ with respect to it. We denote the localized model structure by $M^{(C,S)}$. It can be shown that the fibrant objects in $M^{(C,S)}$ are the simplicial functors $F : C\to M$ that are levelwise fibrant and transfer morphisms in $S$ to equivalences.

Now, let $(F_*C, F_*S)$ be the canonical free resolution of $(C, S)$ in simplicial categories. Then, one has a diagram of pairs of simplicial categories $$(C, S) \xleftarrow{} (F_*C, F_*S)\xrightarrow{}(F_*S)^{−1}(F_*C) = L(C, S),$$ inducing a diagram of right Quillen functors $$M^{(C,S)}\xrightarrow{}M^{(F_*C, F_*S)}\xleftarrow{} M^{L(C,S)}.$$ By Theorem 2.3.5 loc. cit., these right Quillen functors are Quillen equivalences. It follows that two model categories $M^{L(C,S)}$ and $M^{(C,S)}$ are Quillen equivalent. This holds in particular if $C$ is a small (usual) category and $S\subseteq C$ a (usual) subcategory.

In Lurie's Higher Topos Theory it is shown that $M^{L(C,S)}$ is a model for the $\infty$-category of functors from $L(C,S)$ to $M$. It follows that $M^{(C,S)}$ is a model for the $\infty$-category of functors from $(C,S)$ to $M$.