= 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.
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