  We prove that both, the embedding of the category of Hopf algebras into that
of bialgebras and the forgetful functor from the category of Hopf algebras to
the category of algebras, have right adjoints; in other words: every bialgebra
has a Hopf coreflection and on every algebra there exists a cofree Hopf
algebra. In this way we give an affirmative answer to a forty years old problem
posed by Sweedler. On the route the coequalizers and the coproducts in the
category of Hopf algebras are explicitly described.
