Symmetric algebras in categories of corepresentations and smash products

We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra H; for the symmetric property H is assumed to be cosovereign. If H is finite dimensional and A is an H-comodule algebra, we uncover the connection between A   and the smash product A#H⁎A#H⁎ with respect to the Frobenius and symmetric properties.