Ytukyg

At the page 74 of HTT, there is the following theorem

Let $S$ be a simplicial set, $mathcal{C}$ a simplicial category, and $phi: mathfrak{C}[S] rightarrow mathcal{C}^{op}$ a simplicial functor. The straightening and unstraigntening functors determine a Quillen adjunction
$$ St_{phi} : (Set_{Delta})_{/S} leftrightarrows Set_{Delta}^{mathcal{C}} :Un_{phi}$$
where $(Set_{Delta})_{/S}$ is endowed with the contravariant model structure and $Set_{Delta}^{mathcal{C}}$ with the projective model structure. [...]

In then says that the proof is easy, but I can't manage to show that $St_{phi}$ sends cofibrations to projective cofibrations. I thought that since the the class of morphisms which are sent to projective cofibrations is weakly saturated it is enough to show the result for all inclusions $partial Delta^n subseteq Delta^n$.

I did not have much success for the simplicial category $mathcal{C}$ and the map $phi$ could be anything and I have a hard time dealing with it.

Furthermore there is something else which troubles me: the model structure on the $Set^{mathcal{C}}_{Delta}$ makes no use of the simplicial enrichement on both $mathcal{C}$ and $sSet$ so I was wondering if I was not missing something by believing that the that model structure on $Set^{mathcal{C}}_{Delta}$ is really the projective model structure coming from the Kan model structure on $sSet$ and not the one coming somehow from an other model stucture using the simplicial enrichement.