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 $𝔹$ be a monoidal bicategory. An adjoint monoid $M ∈ 𝔹$ is both a 2-monoid $(M,m,u)$ and a 2-comonoid $(M,d,e)$ such that the multiplication is adjoint to the comultiplication, $m \dashv d$ and the unit is adjoint to the counit, $u \dashv e$.