I asked that question in Mathoverflow, 4 years ago, but it was qualified as an off-topic, and they sent me here.
Here's the question: 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.
An even curiouser question is about disjoint union of two (non-small) categories.
The answer to your question depend strictly on the foundational theory you work with.
Assuming your are using a set theory with large enough collections such as NBG or TG, and so you categories are made of classes (and in som cases even sets) objects and classes of morphisms the product categories are build via cartesian products of the classes of objects and morphisms, hence the existance of product categories follows by usual set theoretic axioms.
Clearly the same works for coproduct categories.