Bruennler question

Kai Bruennler asks: 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? The answer is yes if you're willing to use a lot of choice. Perhaps the quickest construction is to assume a well-ordering on the universe with the property that x < y whenever x \in y. Then define the pair l:AxB --> A, r:AxB --> B by stipulating that AxB is a von Neumann ordinal "lexicagraphically ordered" by l and r, that is, (lx < ly) or (lx = ly and rx < ry) whenever (x \in y) and (y \in AxB).