  We investigate the property of being Frobenius for some functors strictly
related with Hopf modules over a bialgebra and how this property reflects on
the latter. In particular, we characterize one-sided Hopf algebras with
anti-(co)multiplicative one-sided antipode as those for which the free Hopf
module functor is Frobenius. As a by-product, this leads us to relate the
property of being an FH-algebra (in the sense of Pareigis) for a given
bialgebra with the property of being Frobenius for certain naturally associated
functors.
