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?
Best Answer
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 $\mathbf{Set}$.