Bergner homotopy category of simplicially enriched caterories is cartesian closed

Question

Let $Cat_{\Delta}$ be the model category of simplicially enriched categories with the Bergner model structure. In a paper I am reading, they state without proof that $Ho(Cat_{\Delta})$ of this model structure is cartesian closed; why is this the case?

Answer 1

The proof I would give is that the Bergner model category is Quillen equivalent to a Cartesian closed model category, and any Cartesian closed model category has a Cartesian closed homotopy category. The most standard such Quillen equivalence is with the Joyal model structure on simplicial sets. It is probably possible to prove this directly, but it seems pretty difficult at a glance.