It is shown that coherence conditions for monoidal categories concerning associativity are analogous to coherence conditions for symmetric strictly monoidal categories, where associativity arrows are identities. Mac Lane’s pentagonal coherence condition for associativity is decomposed into conditions concerning commutativity, among which we have a condition analogous to naturality and a degenerate case of Mac Lane’s hexagonal condition for commutativity. This decomposition is analogous to the derivation of the Yang-Baxter equation from Mac Lane’s hexagon and the naturality of commutativity. The pentagon is reduced to an inductive definition of a kind of commutativity.
"Associativity as commutativity." J. Symbolic Logic 71 (1) 217 - 226, March 2006. https://doi.org/10.2178/jsl/1140641170