Product and coproduct monoidal structures on sets

Cartesian product with singleton unit and disjoint union with empty unit each make the Category of sets monoidal. Associators and unitors simply retag or rebracket the same elements, so the pentagon and triangle identities hold elementwise. These structures differ even up to monoidal equivalence: any category equivalence preserves initial and terminal objects, so it cannot identify the empty unit with the singleton unit.