Talk:Bicategory
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Isn't this what is usually called a (strict) 2-category? (as opposed to weak?) For example, in CFWM, Mac Lane gives quite different definitions for "bicategory" and "2-category". What seems to be described here is actually what is he calls "2-category", and is not the same thing as "bicategory" (to my understanding).
A bicatgeory is a weak 2-category, where associativity holds only up to coherent isomorphism, as stated here.