Once again let me respond to Kai Bruennler's query: Is there a binary product in the category of sets and functions that is "strictly associative", i.e. A x (B x C) = (A x B) x C and the associativity isomorphisms are equal to the identity? With the axiom of choice it's possible to construct such a product for most categories that arise in nature. The needed condition (besides having finite products) is that each isomorphism class be large enough to accommodate the construction in the next sentence. Given a category with a binary products let O be its objects and O* the finite strings of objects and choose a bijection P:O* --> O that sends a string to a product thereof. (The condition, therefore, reduces to the condition that each isomorphism class be at least as large as the class of finite strings whose product is in that isomorphism class.) Define a new binary product on objects by taking P<A_1, A_2,..., A_m> x P<B_1, B_2,..., B_n> = P<A_1, A_2,..., A_m, B_1, B_2,..., B_n>. As written, this construction does not work for the category of sets. The problem is the empty set. The construction, though, continues to work if the bijection condition is understood only for those strings whose products are non-empty. The same trick works for any category with strict initial object. Note that 1 x A = A = A x 1 where 1 denotes P<>.