I'm reading $\textit{The Hopf Ring for Complex Cobordism}$ by Ravenel and Wilson where they discuss the notion of group and ring objects over a category. They say that a hopf algebra over a ring in this context is a group object in the category of coalgebras. Here's my problem. I assume that the group operation for a hopf algebra must be the algebra multiplication, since the addition is presupposed from the coalgebra structure. Not all hopf algebras are division algebras. But, shouldn't a group object have inverses?