As far a I know, there are two kinds of results which are called 'coherence theorems'. For bicategories they take approximately the following forms:
- Every diagram of a certain form commutes
- A Bicategory is equivalent to a 2-catagory.
This second form seems to be more used than the first, however I am wondering what is the general relationship between those two things. Specifically:
- (2) is false when applied to tricategories. Do we know something about (1)?
- (2) also holds for pseudo-fonctors. Does it hold for pseudo-natural transformations? What about (1) ?
Your (1) is usually the key lemma used to prove (2). When you do higher category theory, you will not necessarily have commuting diagrams anymore, though, but they will only commute up to a canonical isomorphism which itself satisfies some axioms.
What do you mean by "(2) is false when applied to tricategories"? Yes, it's false that tricategories are equivalent to strict 3-categories, and there are good reasons for it. One wouldn't expect them to be equivalent. However, there are other coherence theorems for them, for example every tricategory is equivalent to a Gray category, that's a result by Gordon, Power and Street. And I'd expect that the proof goes again like showing that diagrams involving unit laws commute on the nose in the Gray category and those involving interchange of horizontal and vertical composition commute up to some specific isomorphism, the "interchanger".