Alternatively you could require that the product of the products of two sets should be equal to the product of their disjoint union. In particular the product o

Alternatively you could require that the product of the products of two sets should be equal to the product of their disjoint union. In particular the product of any set multiplied with the product of the empty set should remain the same, hence the product of the empty set is the identity. This avoids some problems you get when you try to generalise the categorical the notion of product, as it's not immediately clear (to me anyway) how to get a category where products correspond to an arbitrary group/monoid product.