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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.

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  • $\begingroup$ 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. $\endgroup$
    – Berci
    Jun 13, 2018 at 18:41
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    $\begingroup$ " 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. $\endgroup$
    – Pece
    Jun 14, 2018 at 15:27
  • $\begingroup$ 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. $\endgroup$
    – Vlad Patryshev
    Jun 13, 2019 at 14:56

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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.

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  • $\begingroup$ I read your answer as confirming that we only can build products (and coproducts) in the case where they are provided by some foundation axioms. Nothing like this gives us Set x Set, right? Or Set x Cat. $\endgroup$
    – Vlad Patryshev
    Oct 2, 2018 at 20:52
  • $\begingroup$ @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 $\mathbf{Set}\times\mathbf{Set}$. $\endgroup$
    – Giorgio Mossa
    Oct 3, 2018 at 10:24
  • $\begingroup$ I'm totally confused. How come a ZFC Set is not a category? What is missing there? Sets as object, functions (functional relations) as arrows; id is there, composition is defined and is associative. What's missing? $\endgroup$
    – Vlad Patryshev
    Oct 5, 2018 at 20:31
  • $\begingroup$ Oh, thank you; I could not figure out. Can you edit your comment (and I'm removing mine) $\endgroup$
    – Vlad Patryshev
    Jun 15, 2019 at 22:34
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    $\begingroup$ @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. $\endgroup$
    – Giorgio Mossa
    Jun 16, 2019 at 19:27

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