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.

simplicialfunctors, i.e. the functors that preserve the simplicial enrichment? The projective model structure in question is that of section A.3.3 $\endgroup$ – Denis Nardin Nov 28 '18 at 11:16