Dualizing cartesian and cocartesian fibrations

TL;DR

This paper presents an explicit construction of the dual cocartesian fibration for a given cartesian fibration and demonstrates their classification by the same functor to the infinity-category of categories.

Contribution

It provides a clear and explicit method to construct the dual cocartesian fibration and establishes their classification equivalence.

Findings

Explicit construction of the dual cocartesian fibration.

Dual fibrations are classified by the same functor to Cat_infinity.

Abstract

In this technical note, we proffer a very explicit construction of the "dual cocartesian fibration" $p^{\vee}$ of a cartesian fibration $p$, and we show they are classified by the same functor to $\mathbf{Cat}_{\infty}$.