nLab
comonoid

A comonoid (or comonoid object) in a monoidal category $M$ is a monoid object in the opposite category $M^{\mathrm{op}}$ (which is a monoidal category using the same operation as in $M$).

For example, a comonoid in Vect (with its usual tensor product) is called a coalgebra. Every set can be made into a comonoid in Set (with the cartesian product) in a unique way. More generally, every object in a cartesian monoidal category can be made into a comonoid in a unique way.