I would like to realize $QCat:=(sSet)_{\text{Joyal}}$ as an $(∞,1)-category$ in the sense of Lurie (i.e. weak Kan complex). I would like to do this in the following way:

1) First since $QCat$ is a model category it lives in $RelCat$ the category of small relative categories. So apply the hammock localizaiton and get a category enriched in simplicial sets.

2) Fibrantly replace the category in the Bergner model structure on categories enriched in simplicial sets and take the homotopy coherent nerve to get an $(\infty, 1)$-category.

Question 1: Does this procedure work?

Question 2: Are there some size issues? For example will $QCat$ really be an object of $RelCat$, also if I apply the hammock localization do I get a small simplicial enriched category?

I apologize for these boneheaded questions. (Also, I posted this question on stack exchange but got no answers so I deleted it and am posting here)

interiorof the category and it is sometimes denoted $\iota C$ or $C^\cong$. $\endgroup$ – Denis Nardin May 13 '17 at 2:01