Takeuchi’s free Hopf algebra construction revisited

Takeuchi’s famous free Hopf algebra construction is analyzed from a categorical point of view, and so is the construction of the Hopf envelope of a bialgebra. Both constructions in fact can be described as compositions of well known and natural constructions. This way certain partially wrong perceptions of these constructions are clarified and their mutual relation is made precise. The construction of Hopf envelopes finally is shown to provide a construction of a Hopf coreflection of bialgebras by simple dualization. The results provided hold for any commutative von Neumann regular ring, not only for fields.