In a model category, is the full subcategory of fibrant objects a reflective subcategory? I apologize in advance if my question is utterly stupid, but I can't resist asking it. So...

Is it true that in a model category ( - for example $\mbox{Set}_\Delta$ with the Joyal model structure - ) the full subcategory of fibrant objects is a reflective subcategory? More concretely, is the fibrant replacement functor a left adjoint to the inclusion functor?

Thanks.
Although most model structures appearing "in practice" do not have a reflective subcategory of fibrant objects (unless every object is fibrant), the following result of A. Salch provides many examples that do.

Theorem. Let $\mathcal{C}$ be a complete and cocomplete category, and let $\mathcal{A}$ be a full subcategory which is reflective and replete (i.e., closed under isomorphisms). Then there exists a model structure on $\mathcal{C}$ such that $\mathcal{A}$ consists of the fibrant objects.

See Corollary 6.3 here:

http://arxiv.org/abs/1501.00508v1

The model structure is obtained as a (left) Bousfield localization of the discrete model structure on $\mathcal{C}$. The weak equivalences are the maps that become isomorphisms upon applying the reflector $\mathcal{C} \to \mathcal{A}$.
In the case that all isomorphisms in the model category are fibration, cofibration, and weak equivalences, (trivial cofibrations, fibrations) and (fibrations, trivial cofibrations) form factorization systems in the sense of Borceux: 


  A factorization system in a category $\mathbf B$ is defined as a pair
  $(E, F)$ of classes of arrows in $\mathbf B$ such that
  
  
  every isomorphism belongs to both E and M
  both E and M are closed under composition
  $\forall e \in E \forall m \in M. e \perp m$ (i.e. e has the LLP wrt m)
  every arrow f in B can be factored as f = me with e in E and m in M
  


Then the following theorem in Borceux's Handbook of Categorical Algebra, volume 1, page 211, gives an affirmative answer:


  Suppose $\mathbf B$ is a category with a terminal object 1 and a factorization system (E, F). For each object $B \in \mathbf B$, factor the unique arrow $B \xrightarrow ! 1$ into $B \xrightarrow {e_B} r(B) \xrightarrow {f_B} 1$. Then $r$ is a reflection of $\mathbf B$ into the full subcategory spanned by all $r(B)$, and each $e_B$ is the unit of the reflection.