Remark

It is the fact that this condition makes recourse among all (n,r)-categories to (∞,0)-categories = ∞-groupoids that it is easier to define and handle an (∞,r)-category than an (n,r)-category for finite $n$: in the former case one just needs that certain spaces are contractible without necessarily being equal to the point, while in the latter case one demands some of these to be exactly equal to the point, which is a condition much harder to get under control. Accordingly, a coherence law in an (n,n)-category such as a bicategory ($n = 2$) or a tricategory ($n=3$) or a tetracategory ($n = 4$) is in degree $n+1$ a complicated equation that asserts that a certain contractible space of higher morphisms is exactly equal to the point.

Warning

The demand for contractible spaces of choices of associators and unitors is not to be confused with asserting contractible hom-spaces in general (which would make the theory of $(n,r)$-categories trivial, anyway!)

For instance in a braided monoidal category there is, in general, a non-contractible space of morphisms $x \otimes y \to x \otimes y$ for any two objects, because the double braiding $B_{y \otimes x}\circ B_{x \otimes y}$ will not in general be equal to the identity morphism. But the braiding morphisms also is not the kind of structural morphism that the above definition refers to. In the definition of braided monoidal category it may look as if it is on par with the associator, but this is in fact not so:

in the context of $(n,r)$-categories we may use the periodic table of higher categories to identify the braided monoidal category with a one-object-one-morphism 3-category. In this, the non-trivial double braiding $B_{y \otimes x}\circ B_{x \otimes y}$ is a nontrivial 3-morphism that represents a nontrivial element of the 3rd homotopy group of the 3-category. Demanding this to be trivial would be demanding the 3-category to be trivial!

But the role of the associator, which now is a 3-morphism witnessing the non-associativity of 2-morphisms under horizontal composition is quite different. This is best seen by looking at a geometric definition of higher categories, such as an (∞,n)-category for $n = 3$: here the associator is a choice of horn-filler, and this choice by construction happens in a contractible space and by construction cannot contain nontrivial cells like the double braiding. This is also amplified by the following example