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I have received an email asking for clarification at the (old) entry equivalence of 2-categories, as to the meaning of “essentially full”. I have briefly added a parenthetical “i.e. essentially surjective on hom-categories”. But the entry deserves to be expanded a bit more, maybe somebody feels inspired to do so?
It’s not true in general that a strict 2-functor between strict 2-categories is a biequivalence (i.e. is biessentially surjective on objects and an equivalence on hom-categories) iff it is part of an “equivalence of 2-categories” as currently defined on this page: the ’inverse’ of the 2-functor might only be a pseudofunctor, not a strict 2-functor. See Example 3.1 in Steve Lack’s paper A Quillen model structure for 2-categories.
Feel free to edit the page, it could do with some re-structuring and additions; I was writing quickly earlier just to try to improve things a little. I will adjust the offending paragraph now.
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