The appropriate "pseudo" sense of Frobenius for a pseudomonoid, as mentioned last week, turns out to be exactly what is called "star-autonomous" in [B.J. Day and R. Street, Quantum categories, star autonomy, and quantum groupoids, to appear in "Galois Theory, Hopf Algebras, and Semiabelian Categories", Fields Institute Communications (American Mathematical Society 2004)]. This implies a remarkable connection with quantum groupoids. A weak notion of equivalence, for a morphism in a bicategory, was also defined; in the talk it was called "light equivalence" but currently I am calling it "projective equivalence. Details are available at (link)