  The usual coherence theorem of MacLane for categories with multiplication
assumes that a certain pentagonal diagram commutes in order to conclude that
associativity isomorphisms are well defined in a certain practical sense. The
practical aspects include creating associativity isomorphisms from a given one
by tensoring with the identity on either the right or the left. We show, by
reinspecting MacLane's original arguments, that if tensoring with the identity
is restricted to one side, then the well definedness of constructed
isomorphisms follows from naturality only, with no need of the commutativity of
the pentagonal diagram. This observation was discovered by noting the
resemblance of the usual coherence theorems with certain properties of a
finitely presented group known as Thompson's group F. This paper is to be taken
as an advertisement for this connection.
