  We show that the universal measuring coalgebras between Frobenius algebras
turn the category of Frobenius algebras into a Hopf category (in the sense of
Batista-Caenepeel-Vercruysse), and the universal comeasuring algebras between
Frobenius algebras turn the category Frobenius algebras into a Hopf opcategory.
We also discuss duality and compatibility results between these structures.
  Our theory vastly generalizes the well-known fact that any homomorphism
beween Frobenius algebras is an isomorphism, but also allows to go beyond
classical (iso)morphisms between Frobenius algebras, especially in finite
characteristic, as we show by some explicit examples. The paper is concluded
with some open questions and considerations about Topological Quantum Field
Theories.
