Let me start by considering associativity. Let $C$ be a set with a binary operation (multiplication) $C\times C\to C$. The associativity law is $(ab)c=a(bc)$. It can be expressed by a commutativity law of functions, namely $L_c\circ R_a=R_a\circ L_c$, where $R_a, L_c$ are multiplication on the right/left. Naively, commutativity is simpler than associativity. On the other hand, we have functions other than elements. So associativity of a simpler stuff is express by commutativity of a more complex stuff.
Let us go one step higher, consider the pentagon condition for a binary functor. My question: Is it possible to express the pentagon condition as an associativity law of [blank to seek]? One could ask similar question for more higher associativities.