In Derived Algebraic Geometry, Lurie lets $\mathcal{C}$ be the category of simplicial commutative rings. He remarks that $\mathcal{C}$ admits the structure of a cofibrantly generated model category, and then defines $\mathcal{SCR}$ to be the "corresponding $\infty$-category".
I am trying to understand why we can pass from $\mathcal{C}$ to an $\infty$-category, where by $\infty$-category Lurie means a simplicial set with the weak Kan condition that horns $\Lambda_i^n \to X$ have fillers for $0< i < n$. I will call these quasi-categories.
My best guess is to use the simplicial nerve functor from Higher Topos Theory, page 22, which takes us from simplicially enriched categories to simplicial sets. In order for the nerve to be a quasi-category, we need the simplicial sets $\mathcal{C}(R', R)$ to be Kan complexes. I know that there is a simplicial enrichment of $\mathcal{C}$, defined as follows.
For an object $R\in \mathcal{C}$, define $R^{\Delta[p]} = \textbf{sSet}(\Delta[p], R)$ to be the simplicial set of morphisms $\Delta^p\to R$. This is naturally an object of $\mathcal{C}$, and we define the $p$-simplices of $\mathcal{C}(R', R)$ to be the set of morphisms $R' \to R^{\Delta[p]}$ of simplicial commutative rings. In order to get to a quasi-category using the nerve, I would need the simplicial sets $\mathcal{C}(R', R)$ to be Kan complexes, but I'm not sure if they are.