TITLE: Link between homotopy equivalence of simplicial sets and categorical equivalences
QUESTION [7 upvotes]: In Higher Topos Theory, a map $f: S \rightarrow T$ of simplicial sets is a categorical equivalence if after applying the functor $\mathfrak{C}[-]$ we have an equivalence of simplicial categories. 
In the proof of Theorem 2.2.5.1, we see for example that any trivial fibration, in the Kan model structure on simplicial sets, is a categorical equivalence. 
My question is the following : under what conditions can we say that a Kan weak equivalence is a categorical equivalence? 
I am asking this question because when Lurie finally proves Proposition 1.2.9.3 in section 2.4.5, he claims (I think), at some point  that a weak equivalence between two Kan complexes is a categorical equivalence. (This is to use Corollary 2.4.4.4.)
If more details are needed I will gladly add them.

REPLY [4 votes]: While Dylan's comment is correct, it doesn't really explain how this is shown.  It's actually a corollary of the existence of the Joyal model structure, which is proven earlier in the chapter as a corollary of the comparison theorem with simplicial categories.
Since the cofibrations for both the Joyal and Kan-Quillen model structures coincide, and since the fibrant objects for the Kan-Quillen model structure are also fibrant for the Joyal model structure, it follows that the Kan-Quillen model structure is a left-Bousfield localization of the Joyal model structure.
It is a general fact of the theory of (left) Bousfield localization (see e.g. Hirschhorn) that the local equivalences between local objects are exactly the equivalences in the original unlocalized model structure.
I don't see any way to show this directly just from the given description of the weak equivalences without first proving the comparison theorem.