added the classical recognition theorem for 2-equivalences assuming AC, copied over from the same material that I just wrote into *2-functor* (see the other thread)

As a reference I have added pointer to Thm. 7.4.1 in

- Niles Johnson, Donald Yau, Section 7 of:
*2-Dimensional Categories*, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

but maybe this prompts somobody else to add pointer to original references, if any.

]]>Now done.

]]>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.

]]>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*.

Tweaked things slightly to emphasise that the notion of biequivalence is relevant both when one is working with weak 2-functors and when one is working with strict ones.

]]>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?