It was J.F. Jardine established the so called supercoherence theory in [Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194](http://www.sciencedirect.com/science/article/pii/002240499190122I). The result can be roughly stated as follows:




---


A pseudo-functor $\Delta^{op}\to \mathbf{Cat}$, from the opposite of simplex category to the 2-category of small categories, is fully determined by its value on objects, face maps and degeneracy maps, natural isomorphisms corresponding to the standard simplicial identities, and a small list of higher order coherence conditions (consisting 17 identities).




---


This supercoherence subsumes "all diagrams commute type" coherence for monoidal category, bicategory(?).


Now it's natural to replace the target strict 2-category $\mathbf{Cat}$ by a general bicategory $\mathbf{C}$. Is the naive analogue still true?


The naive approach is first looking at any strict 2-cateogry, then embedding a bicategory in a strict 2-category by the coherence theorem for bicategory. I do not know if this works actually.



What is the suitable setting for supercoherence with value in a bicategory?
As far as I understand, the strictness is not very relevant here since the commutative diagrams are of certain 2-morphisms which are given as data. I don't think that in any place there you need to consider triple compositions of 1-morphisms, so the comparisons simply don't appear. 


