I am little bit lost with the following (standard?) problem in model categories.


Suppose we have a Quillen adjunction between combinatorial model categories:


$$L:M\leftrightarrow N: R $$


and let $(M,Cof,Fib, WE)$ denote the model structure on $M$. I would like to know if it is possible to define a new model structure $(M, Cof, Fib^{'}, WE^{'})$ such that $f \in WE^{'}$ if and only if $L(f)$ is a weak equivalence in $N$ ? 



Localization of a model category with respect to a class of maps
This is a very well-studied problem. Hirschhorn's book proves that, if $M$ is left proper and cellular then localization exists with respect to a set of maps. Jeff Smith figured out how to replace cellular by combinatorial; a nice write-up is in Barwick's paper [On Left and Right Model Categories and Left and Right Localizations](https://projecteuclid.org/download/pdf_1/euclid.hha/1296223884). For a general class of maps, the existence of the localization is conjecturally equivalent to Vopenka's principle. See [The orthogonal subcategory problem in homotopy theory](http://www.ub.edu/topologia/casacuberta/articles/cacho.pdf) by Casacuberta and Chorny (and, also, [Definable orthogonality classes in accessible categories are small](https://arxiv.org/abs/1101.2792)).


However, if the class of maps is *accessible* then Lurie shows how to construct the localization, in Section 5.5 of Higher Topos Theory. Since you start with a left Quillen functor between combinatorial model categories, for you this means $L$ should be an accessible functor and the weak equivalences of $N$ should be of small generation (see Prop 5.5.4.16).


