Let $M$ and $N$ be "nice" model categories. I'm happy to have "nice" mean combinatorial model category. Consider a Quillen pair $$ L: M\rightleftarrows N: R.$$ I want the following result:

There exists a set of maps $S$ in $M$, such that $L$ and $R$ descend to a Quillen pair $$ L: S^{-1}M \rightleftarrows N: R,$$ with the property that a map $f:X\to Y$ between cofibrant objects in $M$ is a weak equivalence in $S^{-1}M$ if and only if $L(f)$ is a weak equivalence in $N$.

Here $S^{-1}M$ denotes the model category with the same underlying category $M$ obtained by localizing $M$ with respect to the set of maps $S$.

This result seems to encode a standard technique; in fact, the very first example of a localized model category (Bousfield's localization of spaces with respect to a homology theory) can be viewed (in retrospect) of a special case of this.

I think I could prove this result if I need to. But I would rather have a reference. I've looked in the usual places, but I can't seem to find anything exactly like it. I expected to find something like this in one of Dugger's papers on presentable model categories, but I don't find it there; he proves that under an additional condition, you can get a Quillen *equivalence* $S^{-1}M\rightleftarrows N$, but his construction of the set $S$ does not apply in my case.