In Higher topos theory and Higher algebra Lurie defines (see section 2.3.1 of HTT) a correspondence between two $\infty$-categories $C$ and $D$ as being an $\infty$-category $\mathcal{M}$ over $\Delta[1]$ whose fiber over $0$ and $1$ are respectively $C$ and $D$. Of course this is supposed to be equivalent to presheaves over $C^{op} \times D$, or to left adjoint functor $\widehat{D} \rightarrow \widehat{C}$. This is vaguely claimed as remark 2.3.1.4 in Higher topos theory, but I do not know if the details of this correspondence have been developped somewhere...


Is there a reference for this equivalence ?
It is possible to deduce it without too much work from other results in the literature ?


I need this results for a technical lemma, and so far I don't see how to prove without some hard work on simplicial sets taking several pages... but I'm hopping I'm missing a nicer argument.


*Remark*: it is easy using Lurie's work to attach to $\mathcal{M} \rightarrow \Delta[1]$ a 'profunctor' $C^{op} \times D \rightarrow$ Space, In fact using the twisted arrow category construction (see section 5.2.1 of Higher algebra) one can even give a very explicit right fibration $T \rightarrow C^{op} \times D$ representing this profunctor, but going the other way seem harder...



Correspondences of $\infty$-categories
Your question is answered by the following result, for which I will give a few references.




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**Theorem.** For each pair of simplicial sets $A$ and $B$, there is a functor $$a\_{A,B}^\* \colon \mathbf{Cyl}(A,B) \longrightarrow \mathbf{sSet}/(A^\mathrm{op} \times B)$$ which is both a left and a right Quillen equivalence between the Joyal model structure on $\mathbf{Cyl}(A,B)$ and the covariant model structure on $\mathbf{sSet}/(A^\mathrm{op} \times B)$. 




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**Definitions.** (1) The category $\mathbf{Cyl}(A,B)$ of *cylinders* from $A$ to $B$ (this is Joyal's terminology) is the fibre over the pair $(A,B)$ of the functor $$\mathbf{sSet}/\Delta[1] \longrightarrow \mathbf{sSet} \times \mathbf{sSet}$$ that sends a map $X \to \Delta[1]$ to its fibres over $0$ and $1$. One can show that the category $\mathbf{Cyl}(A,B)$ admits a model structure (called the "Joyal model structure" above) created from Joyal's model structure for quasi-categories by the forgetful functor $$\mathbf{Cyl}(A,B) \longrightarrow \mathbf{sSet}/\Delta[1] \longrightarrow \mathbf{sSet}.$$ If $A$ and $B$ are quasi-categories, then the fibrant objects of this model structure are precisely the correspondences from $A$ to $B$ as you have defined them.


(2) The functor $a\_{A,B}^\*$ sends a cylinder $X \in \mathbf{Cyl}(A,B)$ to the pullback
$\require{AMScd}$
\begin{CD}
 a\_{A,B}^\*(X) @>>> \mathrm{Tw}(X)\\
 @V V V @VV V\\
 A^\mathrm{op} \times B @>>i^\mathrm{op}\times j> X^\mathrm{op} \times X
\end{CD}
where $\mathrm{Tw}(X)$ denotes the twisted arrow of $X$ (whose $n$-simplices are maps $\Delta[n]^\mathrm{op} \star \Delta[n] \to X$), and where $i \colon A \to X$ and $j \colon B \to X$ denote the inclusions of the fibres of the structure map $X \to \Delta[1]$.




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**Proofs.** In the case where $A$ and $B$ are quasi-categories, the theorem above is proved by Danny Stevenson in his recent preprint (see Theorem C therein)



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> Danny Stevenson. *Model structures for correspondences and bifibrations*. [arXiv:1807.08226](https://arxiv.org/abs/1807.08226).
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(Note that Stevenson uses the term "correspondence" for what I have called "cylinder".)


As I shall explain below, the general case of the theorem can be proved by a combination of results from my recent preprint 



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> Alexander Campbell. *Joyal's cylinder conjecture*. [arXiv:1911.02631](https://arxiv.org/abs/1911.02631).
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and from Cisinski's recent book



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> Denis-Charles Cisinski. *Higher categories and homotopical algebra*, vol. 180 of *Cambridge studies in advanced mathematics*. Cambridge University Press, 2019.
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First, note that the category $\mathbf{Cyl}(A,B)$ is equivalent to the category $\mathbf{ssSet}/(A^\mathrm{op} \boxtimes B)$ of bisimplicial sets over the exterior product of $A^\mathrm{op}$ and $B$. It follows from Remark 3.23 and Theorem 5.4 of my preprint that, under this equivalence, the Joyal model structure on $\mathbf{Cyl}(A,B)$ corresponds to the *bicovariant model structure* on $\mathbf{ssSet}/(A^\mathrm{op} \boxtimes B)$ defined in Section 5.5 of Cisinski's book. 


Now, it is not difficult to see that, under this equivalence, the functor $a\_{A,B}^\*$ defined above corresponds to the functor $$\delta\_{A^\mathrm{op},B}^\* \colon \mathbf{ssSet}/(A^\mathrm{op} \boxtimes B) \longrightarrow \mathbf{sSet}/(A^\mathrm{op} \times B)$$ which sends a bisimplicial set over $A^\mathrm{op} \boxtimes B$ to its diagonal (note that the diagonal of $A^\mathrm{op} \boxtimes B$ is $A^\mathrm{op} \times B$).


Hence the theorem follows from Theorem 5.5.24 in Cisinski's book, where he proves that the functor $\delta\_{A^\mathrm{op},B}^\*$ is both a left and a right Quillen equivalence between the bicovariant model structure on $\mathbf{ssSet}/(A^\mathrm{op} \boxtimes B)$ and the covariant model structure on $\mathbf{sSet}/(A^\mathrm{op} \times B)$.


