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.



Infinity category of functors from a relative category to a model category
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 $M$.


