Question. It is widely known that strict 2-categories "are" bicategories for which all coherence 2-cells are identities. However, how to make the "are" precise here?
Remarks.
- I have seen experienced authors resort to circumlocutions when getting to this point. And I found myself resort to such a circumlocution recently.
- A very useful philosophy of category theory is of course that only such comparisons make sense which happen within the same category. So one would wish for a category which contains both "strict 2-categories" and "bicategories for which all coherence 2-cells are identities" as objects. However, I do not know of any such category. In particular, it would have to be a category which does not have 2-categories for its objects, rather it would have to have "axiomatizations of 2-categories" for its objects.
- The "syntactic category" in the sense of categorical logic is vaguely related in spirit, but does not help with the question.
- Model-theoretically speaking, the classes "strict 2-categories" and "bicategories for which all coherence 2-cells are identities" are customarily axiomatized over different signatures.
I can't see the problem.
Even in a bicategory you have the vertical composition that is strict, meaning that the 2-cells with vertical composition form a category, in which the various axioms hold strictly and whose objects are the 1-cells.
Coherence morphisms for bicategory are special 2-cells between different composite 1-cells. In a strict 2-category you require that these coherence 2-cells are identities for the vertical composition which, allow me to stress it again, it's an ordinary 1-category composition (hence strictly associative and unital).
Hope this helps.