Let $(C, \otimes)$ be an exact monoidal category, and let $H(C)$ be the category of cocommutative and commutative Hopf monoids in $C$ with respect to $\otimes$.
What can be said about this category? Is it always abelian? If so, are there always enough projectives or injectives? I know that if $C$ is the category of vector spaces over some field with the standard tensor, then $H(C)$ is abelian. Can this be generalized?