*Digression*: interesting stack overflow post from subsequent googling: https://math.stackexchange.com/questions/175169/direct-products-in-the-category-rel : the concise argument in the first answer shows that while the monoidal product in Rel is the cartesian product, the categorical product is union. If I'm not mistaken, John has elsewhere remarked on the importance of this fact for more sophisticated categories than Rel. But I found the concision of the proof in the link illuminating.