In *Higher Topos Theory*, a map $f: S \rightarrow T$ of simplicial set is a categorical equivalence if after applying the functor $\mathfrak{C}[-]$ we have a 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 condition can we say that a Kan weak equivalence is a categorical equivalence? 

I am asking this question because when Lurie finally proves Propisition 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 equivalences. (This is to use Corollary 2.4.4.4.)

If more details are needed I will gladly add them.