Your question is answered by the following result, for which I will give a few references.

**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 $\mathbf{sSet}/(A^\mathrm{op} \times B)$.

**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 $X \to \Delta[1]$.

**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)

Danny Stevenson. *Model structures for correspondences and bifibrations*. arXiv:1807.08226.

(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

Alexander Campbell. *Joyal's cylinder conjecture*. arXiv:1911.02631.

and from Cisinski's recent book

Denis-Charles Cisinski. *Higher categories and homotopical algebra*, vol. 180 of *Cambridge studies in advanced mathematics*. Cambridge University Press, 2019.

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 $\mathbf{sSet}/(A^\mathrm{op} \times B)$.