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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.

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    $\begingroup$ I agree that it's not easy. Much of chapters 2-4 has been written in easier to read ways since HTT came out. I'd take a look at Moerdijk and Heuts for a more complete argument: arxiv.org/pdf/1308.0704.pdf $\endgroup$ – Kevin Arlin Nov 25 '18 at 17:36
  • $\begingroup$ Thank you for the link, I will take a look at it. However, Lurie seems to have a relatively simple argument in mind and I was really wondering what it could be. $\endgroup$ – Oscar P. Nov 26 '18 at 10:25
  • $\begingroup$ You might find someone to answer this at MO. $\endgroup$ – Kevin Arlin Nov 26 '18 at 18:59
  • $\begingroup$ Cross-posted: mathoverflow.net/questions/316393 $\endgroup$ – Watson Nov 28 '18 at 10:37

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