Given a simplicial model category $\mathcal{M}$ and a set $S$ of morphisms in it, we can define the concept of $S$-local objects. In nLab page (below Definition 3.2), a fibrant object $X\in \mathrm{Ob}(\mathcal{M})$ is $S$-local if for any map $f:A\rightarrow B$ in $S$, the induced morphism $\mathbb{R}f^*:\mathbb{R}\mathrm{Map}(B,X)\rightarrow \mathbb{R}\mathrm{Map}(A,X)$ between simplicial sets is an isomorphism in $\mathrm{Ho}(sSet)$, which is also equivalent to say $Qf^*:\mathrm{Map}(QB,X)\rightarrow \mathrm{Map}(QA,X)$ is a weak equivalence in $sSet$ where $QA$ and $QB$ are cofibrant replacement. It's also the definition accepted by Clark Barwick in the paper On (Enriched) Left Bousfield Localization of Model Categories (Definition 2.6).
But in the book Model Categories and Their Localizations of Philip S. Hirschhorn (Definition 3.1.4), he requires $f^*:\mathrm{Map}(B,X)\rightarrow \mathrm{Map}(A,X)$ is a weak equivalence and do not pass to cofibrant replacements. I want to know whether the two definitions are equivalent.
When I try to prove the equivalence, it seems I need to prove for a trivial fibration $p:QA\rightarrow A$ where $QA$ is cofibrant, the map $p^*:\mathrm{Map}(A,X)\rightarrow \mathrm{Map}(QA,X)$ is a weak equivalence. But I can not prove it.
The two Map's are different (and have different notations in the respective sources).
nLab's Map (denoted by $C(-,-)$ there) is the hom-object given by the simplicial enrichment of the category $C$. In particular, $\def\rdf{{\bf R}}\def\Map{{\rm Map}}\rdf\Map$ is the derived hom-object functor.
Hirschhorn's Map (denoted by $\def\map{{\rm map}}\map$ there) is the homotopy function complex defined using Reedy (co)fibrant resolutions of the source and/or target.
The very point of simplicial model categories is that $\rdf\Map$ and $\map$ are weakly equivalent functors. See Example 17.1.4 in Hirschhorn's book and references therein.
Thus, the two definitions of left Bousfield localizations are equivalent.