Uniqueness of monoidal adjunctions

TL;DR

This paper investigates the uniqueness of certain dual equivalences between categories of monoidal $$-categories, showing that two natural equivalences with compatible structures are canonically the same.

Contribution

It establishes the canonical equivalence of two dual $$-category equivalences equipped with compatible monoidal presheaf functors.

Findings

Two dual equivalences are shown to be canonically equivalent.

The space of equivalences between these categories is analyzed.

Compatibility with monoidal presheaf functors is crucial for the equivalence.

Abstract

There are two dual equivalences between the $\infty$-category of $\mathcal{O}$-monoidal $\infty$-categories with right adjoint lax $\mathcal{O}$-monoidal functors and that with left adjoint oplax $\mathcal{O}$-monoidal functors, where $\mathcal{O}$ is an $\infty$-operad. We study the space of equivalences between these two $\infty$-categories, and show that the two equivalences equipped with compatible $\mathcal{O}$-monoidal presheaf functors are canonically equivalent.