Duoidal $\mathrm{\infty }$-categories of operadic modules

In this paper we study duoidal structures on $\mathrm{\infty }$-categories of operadic modules. Let ${\mathcal{O}}^{\otimes }$ be a small coherent $\mathrm{\infty }$-operad and let ${\mathcal{P}}^{\otimes }$ be an $\mathrm{\infty }$-operad. If a $\mathcal{P}\otimes \mathcal{O}$-monoidal $\mathrm{\infty }$-category ${\mathcal{C}}^{\otimes }$ has a sufficient supply of colimits, then we show that the $\mathrm{\infty }$-category ${\mathrm{M}\mathrm{o}\mathrm{d}}_A^{\mathcal{O}}(\mathcal{C})$ of $\mathcal{O}$-$A$-modules in ${\mathcal{C}}^{\otimes }$ has a structure of $(\mathcal{P},\mathcal{O})$-duoidal $\mathrm{\infty }$-category for any $\mathcal{P}\otimes \mathcal{O}$-algebra object $A$.