# products and coproducts in the category of all categories

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.

• We can always talk about the categorial products and coproducts, at most some don't exist. However, I guess, these will (implicitly assumed to) exist in any foundational framework. Jun 13, 2018 at 18:41
• " keeping in mind that category theory is not based on sets at all" : this is where you are wrong. Most of the time, mathematicians build category theory on top of some set theory, at least a naive one where forming pairs is completely allowed. Alternative foundations, as dependent type theory, will allow pairs as well.
– Pece
Jun 14, 2018 at 15:27
• Assuming that any theory is based on a set theory leads to a question whether a set theory is based on a set theory. And then bumping right into Curry's paradox. Jun 13, 2019 at 14:56

• @VladPatryshev I would say yes, but consider that for instance in ZFC $\mathbf{Set}$ is not a category, so it would be meaningless to talk about things such $\mathbf{Set}\times\mathbf{Set}$. Oct 3, 2018 at 10:24
• @VladPatryshevVlad $\mathbb{Set}$ is a locally small category, meaning that its collection of objects is not a set, thus Set it cannot be a category inside ZFC, exactly like $\mathbb{Set} \times \mathbb{Set}$ or the other examples you're interested in. Jun 16, 2019 at 19:27