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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.

First, notice that if $Xhookrightarrow Y$ is an injective map over $S$, then the map $M_{X,phi} to M_{Y,phi}$ is a cofibration of simplicial categories. To see this, notice that it is a pushout of the map $M_{X,id}to M_{Y,id},$ which can easily be verified to be a cofibration, as it is the image of the injective map $$X^trianglerightcoprod_X S hookrightarrow Y^triangleright coprod_Y S$$ under the functor $mathfrak{C},$ which preserves cofibrations.

Then we see that if $Cto D$ is a cofibration of simplicial categories, then given any $xin C$ such that $C(-,x)$ is projectively cofibrant, the induced natural transformation $C(-, x)to D(f-,fx)$ is a projective cofibration by A.3.3.9.ii.

The last thing that remains to be proved is that $M_{phi,X}(-,p),$ where $p$ denotes the image of the cone point, is projectively cofibrant. The previous revision to this answer did not answer this question, and I am leaving it open until I have more time later or until someone adds an answer.

I think what Lurie might have meant when he wrote "It is easy to see that $St_{phi}$ preserves cofibrations" in the proof of Theorem 2.2.1.2, is that it is easy to see it if you take into account the compatibility of straightening with left Kan extensions as described just above in Proposition 2.2.1.1. Indeed, combining (1) and (2) of the latter proposition we see that given any generating cofibration $partial Delta^n to Delta^n to S$ in $({rm Set}_{Delta})_{/S}$, in order to show that $St_{phi}(partial Delta^n to Delta^n)$ is a projective cofibration in ${rm Set}_{Delta}^{{cal C}}$ it is enough to show that $St_{Id}(partial Delta^n to Delta^n)$ is a projective cofibration in ${rm Set}_{Delta}^{mathfrak{C}[Delta^n]^{op}}$. The map $St_{Id}(partial Delta^n to Delta^n)$ can then be described very explicitly, and one can check that it is a pushout of a map of the form $(i_0)_!partial ((Delta^1)^n) to (i_0)_!(Delta^1)^n$, where $(i_0)_!$ denotes the left Kan extension functor along the terminal object inclusion ${0} subseteq mathfrak{C}[Delta^n]^{op}$ and $partial ((Delta^1)^n) to (Delta^1)^n$ is the inclusion of the boundary of the $n$-cube inside the full $n$-cube.