Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 10 / ii / Solution

The Category of sets is monoidal under Cartesian product, with any singleton as unit. The tensor of maps is $(f\times g)(x,y)=(f(x),g(y))$, the associator sends $((x,y),z)$ to $(x,(y,z))$, and the left and right unitors discard the singleton coordinate. These are natural bijections. Both pentagon paths send a nested quadruple to the same rebracketed quadruple, and both triangle paths discard the same singleton coordinate, verifying the axioms directly.

This is not the only structure. Disjoint union $X⨿Y$ is another tensor bifunctor, with empty-set unit. Its associator changes parentheses while retaining the tag indicating the original summand, and its unitors remove an empty summand. Every coherence path preserves both the original tag and the element, so again the pentagon and triangle hold.

These product and coproduct monoidal structures on sets are distinct even up to monoidal equivalence. Any equivalence of the underlying category preserves initial and terminal objects up to isomorphism. It therefore cannot carry the empty-set unit of the coproduct structure to the singleton unit of the product structure. Hence Set has at least these two inequivalent monoidal structures.