Uniqueness of monoidal adjunctions

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

Keywords

monoidal $\infty$-category, lax monoidal functor, Day convolution, $\infty$-operad

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The author was partially supported by JSPS KAKENHI Grant Number JP17K05253.

Received 13 September 2023

Received revised 23 January 2024

Accepted 28 January 2024

Published 9 October 2024