Let $\mathcal{M}$ be a model category with all nice properties (simplicial, proper, cellular etc...) Let $S$ be a set of maps in $\mathcal{M}$ and let $\mathrm{L}_S\mathcal{M}$ denote the left Bousfield localization of $\mathcal{M}$ with respect to $S.$
Recall that a $S$-local object $W$ is a fibrant object of $\mathcal{M}$ such that for every map $g:X\longrightarrow Y$ in $S$ the induced map $$g^*: \underline{\mathcal{M}}(Y,Z)\longrightarrow \underline{\mathcal{M}}(X,Z) $$ is a weak equivalence. The $S$-local objects are precisely the fibrant objects of the localization $\mathrm{L}_S\mathcal{M}$
Suppose that we have identified a set of maps $J^{local}$, such that an object $Z$ is $S$-local if and only if $Z\longrightarrow *$ has the right lifting property with respect to $J^{local}$. How to proceed from this to prove that a map $Z\longrightarrow W$ is a fibration in $\mathrm{L}_S\mathcal{M}$ if and only if it has the right lifting property with respect this set of maps, that is, $J^{local}$ ? Can it be done at all?