On epimorphisms and monomorphisms of Hopf algebras

We provide examples of non-surjective epimorphisms H → K in the category of Hopf algebras over a field, even with the additional requirement that K have bijective antipode, by showing that the universal map from a Hopf algebra to its enveloping Hopf algebra with bijective antipode is an epimorphism in HopfAlg, although it is known that it need not be surjective. Dual results are obtained for the problem of whether monomorphisms in the category of Hopf algebras are necessarily injective. We also notice that these are automatically examples of non-faithfully flat and respectively non-faithfully coflat maps of Hopf algebras.