Map pseudomonoid

Definition

In a monoidal bicategory, a monoid and a comonoid may be related by two adjunctions. An adjoint monoid is a monoid-comonoid pair where the multiplication is left adjoint to the comultiplication and the unit is left adjoint to the counit.

Definition. Let $\mathbb{B}$ be a monoidal bicategory. An adjoint monoid $M\in \mathbb{B}$ is both a 2-monoid $(M,m,u)$ and a 2-comonoid $(M,d,e)$ such that the multiplication is adjoint to the comultiplication, $m⊣d$ and the unit is adjoint to the counit, $u⊣e$.

Tags: pseudomonoid, Handbook of Categorical Algebra (Borceux). #aug21