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In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between peseudo functors into the category of groupoids and functors cofibered in groupoids.

This leads to my question: why is the following theorem which can be found in Lurie's HTT (Theorem 2.1.2.2) an $\infty$-categorical analogue?

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 $$ St_{\phi} : (Set_{\Delta})_{/S} \rightleftarrows Set_{\Delta}^{\mathcal{C}} :Un_{\phi}$$ where $(Set_{\Delta})_{/S}$ is endowed with the contravariant model structure and $Set_{\Delta}^{\mathcal{C}}$ with the projective model structure. Furthermore, if $\phi$ is an equivalence of simplicial categories, then it is an Quillen equivalence.

I know that fibrant object in the category of objects over $S$ (with the covariant model structure) are right fibrations- the $\infty$-categorical analogue of functors fibered in groupoids- and that the projectively fibrant functors are pointwise Kan complexes -the $\infty$-categorical analogue of groupoid. How does the theorem link those to two togethers?

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  • $\begingroup$ Take $S$ to be an $\infty$-category. Then the theorem produces an equivalence of $\infty$-categories between the $\infty$-category $\mathsf{RFib}(S)$ of right fibrations over $S$ and the $\infty$-category of functors $S^{op} \to \mathsf{Spaces}$. (Take $\mathcal{C} = \mathfrak{C}[S]^{op}$). So the "link" is that they are equivalent. $\endgroup$ – Dylan Wilson Feb 8 at 17:00
  • $\begingroup$ The projective model structure (and also the injective model structure...) on simplicial functors models the $\infty$-category of functors between both sides- see HTT.4.2.4.4. $\endgroup$ – Dylan Wilson Feb 8 at 17:03

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