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.

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.