Endocells of a map pseudomonoid are duoidal

Proposition. Let $(\mathbb{A},m,u,d,e)$ be an adjoint pseudomonoid. The endomaps, $\mathrm{End}(\mathbb{A})$, form a duoidal category with composition and convolution.

Example. In the monoidal bicategory of profunctors, any monoidal category $(\mathbb{A},\otimes ,I)$ induces a duoidal structure on its endoprofunctors, given by composition and Day convolution.

Tags: Map pseudomonoid, duoidal category, compositional dependencies with duoidals.

This results appears in Section 9.1 of Commutativity (Garner, Lopez Franco, 2015), which in turn cites Proposition 4 of Monoidal Bicategories and Hopf Algebroids (Day, Street, 1997).