# Quillen equivalence between sSet (Joyal's model structure) and sSetCat (Bergner's one)

Let me consider two model categories:

• $\mathsf{sSet}$: the category of simplicial sets with Joyal model structure,
• $\mathsf{sSetCat}$: the category of simplicially enriched categories with Bergner model structure.

In Lurie's "Higher Topos Theory", he showed there is a Quillen equivalence between them. Lurie denote it as:

• $\operatorname{N}: \mathsf{sSetCat} \to \mathsf{sSet}$: homotopy coherent nerve,
• $\mathfrak{C}: \mathsf{sSet} \to \mathsf{sSetCat}$.

However, his proof is not easy to follow I think, because of so many hyperlinks.

Is there another proof on this fact, or a document which explains the outline of his proof?