equivalences in bicategories

In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. two counterexamples are provided for that goal, and detailed proofs are given. in particular, all calculations done in a bicategory are fully explicit, in order to overcome the difficulties which arise when working with bicategories instead of 2-categories.