On the straightening of every functor

TL;DR

This paper proves that any functor between ∞-categories can be 'straightened' by establishing an equivalence with lax functors into the correspondence double ∞-category, leveraging a universal property of the Morita category.

Contribution

It establishes a universal straightening equivalence for functors between ∞-categories, connecting overcategories with lax functors into the correspondence ∞-category.

Findings

Proves the equivalence between overcategories and lax functors into Corr.

Uses a universal property of the Morita category in the proof.

Provides a framework for understanding functor straightening in ∞-categories.

Abstract

We show that any functor between $\infty$-categories can be straightened. More precisely, we show that for any $\infty$-category $\mathcal{C}$, there is an equivalence between the $\infty$-category $(\mathrm{Cat}_{\infty})_{/\mathcal{C}}$ of $\infty$-categories over $\mathcal{C}$ and the $\infty$-category of unital lax functors from $\mathcal{C}$ to the double $\infty$-category $\mathrm{Corr}$ of correspondences. The proof relies on a certain universal property of the Morita category which is of independent interest.