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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.

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    $\begingroup$ Category theory is very much based on sets! Who told you otherwise? $\endgroup$
    – Eric Wofsey
    Jun 14, 2014 at 6:20
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    $\begingroup$ There is a notion of product in a bicategory, it is defined similarly to products in categories, but with 2-morphisms implementing higher coherence data. The bicategory of all categories admits all products and coproducts (and in fact all limits and colimits). Before Lawvere characterized the category of sets as a well-pointed elementary topos with a natural numbers objects he tried to characterize the category of all categories in a similar way, though as far as I understand his classification did not succeed because he used categories instead of bicategories. $\endgroup$
    – Dmitri Pavlov
    Jun 14, 2014 at 6:22
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    $\begingroup$ @EricWofsey not all formulations of category theory require sets. $\endgroup$
    – Tim Seguine
    Jun 14, 2014 at 11:10
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    $\begingroup$ Eric Wofsey, category theory is not based on sets. Or else there would be a different category theory for each set theory. Btw, I hope you do not imply set theory is based on set theory. $\endgroup$
    – Vlad Patryshev
    Jun 14, 2014 at 13:23
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    $\begingroup$ @VladPatryshev: I have no idea what your comment means. I also don't really know what your question means. What do you mean when you say "in category theory"? When most people talk about category theory, category theory isn't something that just stands alone; it's formulated within some foundation for mathematics just like any other part of math. Pretty much any foundation for mathematics involves an axiomatization of something that can reasonably be called "sets". $\endgroup$
    – Eric Wofsey
    Jun 14, 2014 at 20:18

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My copy of Mac Lane's book is a few thousand miles away from me at the moment, so the following is based on possibly faulty memory, but here goes anyway: Mac Lane works in a set-theoretic foundational system, something like ZFC plus one Grothendieck universe. So he has no problem with the existence of ordered pairs.

There are people who want to use categories as a foundational system, without any reliance on set theory. These people need to design their foundational system so as to ensure the existence of products of categories and lots of other things that would ordinarily be provided by a set-based foundation.

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