Endocells of a map pseudomonoid are duoidal
Proposition. Let $(𝔸,m,u,d,e)$ be an adjoint pseudomonoid. The endomaps, $\operatorname{End}(𝔸)$, form a duoidal category with composition and convolution.
Example. In the monoidal bicategory of profunctors, any monoidal category $(𝔸,⊗,I)$ 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).