Correspondences between simplicial sets (and $\infty$-categories) are a generalization of the notion of profunctor between categories. It is known that functors between categories are classified by lax diagram of profunctors. We will present this fact from the lens of double category theory. Then, we will show how simplicial sets, simplicial maps and correspondences are organized in a simplicial category (this is a weak simplicial object in categories). A simplicial category may be regarded as a 2-fold version of a simplicially enriched category, and hence some ideas from double category theory apply. In particular we formulate the fact that simplicial maps are classified by diagrams of correspondences. As a corollary, we obtain a formulation of Lurie's prediction that inner fibrations are classified by diagrams of correspondences between $\infty$-categories.
Reference: https://arxiv.org/abs/2005.11597