Let $\mathcal{M}$ be a model category and $S$ a set of cofibrations between cofibrant objects. Then every $S$-local object has the right lifting property with respect to $S$. The converse does not hold in general, but it does under suitable closure conditions on $S$. I think the following should be enough:
Proposition: Suppose that, for every map $f : A \to B$ in $S$, there are cylinder objects $C(A)$ and $C(B)$ and a morphism of cylinder objects $C(A) \to C(B)$ such that the map $C(A) \amalg_{(A \amalg A)} (B \amalg B) \to C(B)$ belongs to $\mathrm{cof}(J \cup S)$, where $J$ is the class of trivial cofibrations of $\mathcal{M}$. Then a fibrant object of $\mathcal{M}$ is $S$-local if and only if it has the right lifting property with respect to $S$.
I can propably prove this proposition myself, but I'd prefer not to write down the proof. So, I'd like to know whether such a proposition appears in the literature (I could not find such a result in Hirschhorn's book). Alternatively, I will accept a proof of this proposition if it is short and slick.