Uniqueness of monoidal adjunctions

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