nLab
comodule algebra

Given a bimonoid $B$ and a monoid $A$ in a symmetric monoidal category $C$, a right coaction $\rho :A\to A\otimes B$ is a right Hopf coaction if $\rho$ is a morphism of monoids where $A\otimes B$ has the standard structure of a tensor product of monoids in the symmetric monoidal category $C$.