Two model structures of some category

Question

Let $\mathscr{C}$ be a category and $(W,C,F)=$(weak equivalence,cofibration,fibration) is some model structure of $\mathscr{C}$. Another model structure of $\mathscr{C}$, $(W_{S},C,F_{S})$ are called stronger if $W \subset W_{S}$. $Ex$ and $Ex_{S}$ are fibrant replacement functors in each model structures. When $X\longrightarrow Y$ belongs to $W_{S}$, $Ex_{S}(X)\longrightarrow Ex_{S}(Y)$ belong to $W$. Since $F=RLP(C\cap W)$ and $W \subset W_{S}$, $F_{S}\subset F$.

In my situation, $\mathscr{C}$ is the category of pointed or unpointed simplicial sets, $(W,C,F)$ is ordinary model structure and $(W_{S},C,F_{S})$ is its localization by some monomorphism $f\colon A\longrightarrow B$ and $W_{S}=\{f$-local equivalence} and $F_{S}=RLP(C\cap W_{S})$.

Q. Is $Ex_{S}(F) \subset F_{S}$ ? In other words, when $X\longrightarrow Y$ is a fibration, is $Ex_{S}(X)\longrightarrow Ex_{S}(Y)$ a stronger fibration?

Answer 1

I think the answer in general is no, i.e. for an arbitrary fibrant replacement functor $Ex_S$ in $M_S = (W_S,C,F_S)$ you will not have this property. However, you can always choose a fibrant replacement functor $T$ that does have this property, because for any $f: X\to Y$, $Ex_S(f)$ can be factored in $M_S$ into a trivial cofibration followed by a fibration (between fibrant objects of $M_S$), and you can define $T(f)$ to be the fibration in that factorization.