It is known that monoidal functors between internal groupoids in the\ud category Grp of groups constitute the bicategory of fractions of the\ud 2-category Grpd(Grp) of internal groupoids, internal functors and\ud internal natural transformations in Grp, with respect to weak\ud equivalences (that is, internal functors which are internally fully\ud faithful and essentially surjective on objects). Monoidal functors can\ud be equivalently described by a kind of weak morphisms introduced by B.\ud Noohi under the name of butterflies. In order to internalize monoidal\ud functors in a wide context, we introduce the notion of internal\ud butterflies between internal crossed modules in a semi-abelian category\ud C, and we show that they are morphisms of a bicategory B(C). Our main\ud result states that, when in C the notions of Huq commutator and Smith\ud commutator coincide, then the bicategory B(C) of internal butterflies is\ud the bicategory of fractions of Grpd(C) with respect to weak\ud equivalences. (C) 2013 Elsevier Inc. All rights reserved

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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.