  It is shown that coherence conditions for monoidal categories concerning
associativity are analogous to coherence conditions for symmetric or braided
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.
