Quantum categories were introduced in [4] as generalizations of both
bi(co)algebroids and small categories. We clarify details of that work. In
particular, we show explicitly how the monadic definition of a quantum category
unpacks to a set of axioms close to the definitions of a bialgebroid in the
Hopf algebraic literature. We define notions of functor and natural
transformation for quantum categories.Comment: Revised and expanded. A lot of diagrams. 40 page

Chikhladze, Dimitri