MacLane, as well as probably any other category book, does not hesitate to define a product of two categories as a category consisting of pairs of objects, etc.

Now my question is: what law of nature or logic or anything allows to create such pairs? Pair creation may be an axiom, say, in Set Theory. In category theory there's no such thing; they seem to just fall from heaven, keeping in mind that category theory is not based on sets at all. It looks pretty suspicious to me; but maybe I'm wrong.

Any comments?

P.S. An even curiouser question is about disjoint union of two (non-small) categories.