# How to think about the Cartesian product in category theory?

I'm beginning to study category theory, and I encountered this result:

In any category with a terminal object 1, any object X is itself a Cartesian product of X and 1. (Halvorson, p. 34)

This certainly clashes with the set-theoretic view of the Cartesian product of two objects (sets in that case) being an entity of a higher rank, and thus non-identical with either of these objects. So, my question is: how to think about Cartesian products in the category theoretic setting?

• They mean that $X$ is a product of $X$ and $1$, and this is trivial to check using the definition of a product (en.wikipedia.org/wiki/Product_(category_theory)). Don't confuse it with the cartesian product of sets, which provides a product in $\mathbf{Set}$. Aug 31, 2021 at 10:40
• @MartinBrandenburg Thanks, that's clear now.
– fr_
Aug 31, 2021 at 10:46
• Sorry I should not post answers as comments, in particular when I always complain about that when others do it :D. I copied the comment to an answer. Aug 31, 2021 at 11:57

They mean that $$X$$ is a product of $$X$$ and $$1$$, and this is trivial to check using the definition of a product. Don't confuse it with the cartesian product of sets, which provides a (possibly different, but isomorphic) product in $$\mathbf{Set}$$.
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