I've just been introduced to category theory. I understand the basic definitions, and I'm trying to get some intuition on how categories tick. I'm wondering: is there a category $C$ such that:
- Its objects are either the integers, the positive integers, or the nonnegative integers (I'm fine with any one of these)
- Its category-theoretic product is the same as the usual product
- Its coproduct is the same as the usual addition?
Intuitively, I think the answer is yes, but I can't come up with a construction.