My first question in this forum. Suppose we have a Quillen equivalence $L: M\leftrightarrow N: R$ between two model categories (say left proper cofibrantly generated cellular or locally presentable). Let $S$ a set of morphisms in $M$ and $L(S)=T$ the image of S in $N$.

I was wondering if $L: \mathbf{L}_{S}M\leftrightarrow \mathbf{L}_{T}N: R$ is a Quillen equivalence? $\mathbf{L}_{S}M$ is the left Bousfield localization of $M$ at $S$ and $\mathbf{L}_{T}N$ is the left Bousfield localization of $N$ at $T=L(S)$

**Remark:** $L(S)$ is the ordinary image **not** the derived one (as in David White answer), there is no cofibrant replacement.

**Edit**: I was wondering if there is a clear answer to my question ? The only reference I have found is here, page 5017 just after theorem 5.7 But I'm still not satisfied since there is no proof and the cited version uses Hirschhorn book where the author (Hirschhorn) deals with the derived image. I will be happy for any clear answer, thank you.