nLab
pseudomonoid

A pseudomonoid in a monoidal 2-category is a categorification of the notion of a monoid object in a monoidal category.

A pseudomonoid in the cartesian monoidal 2-category Cat is precisely a monoidal category. The general definition can be extracted from this special case in a straightforward way.

Just as a monoid in a monoidal category $C$ can be equivalently defined as a monad in the corresponding one-object 2-category $BC$ (the delooping of $C$), so a pseudomonoid in a monoidal 2-category $C$ can equivalently be defined as a pseudomonad in the corresponding one-object 3-category $BC$.