Category on integers with the usual product and coproduct?

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.

Solution 1:

For the non-negative integers, take the category of finite sets. (The exponential here is also the same as the usual exponential.)

For the integers, the answer is no. More generally, the category-theoretic coproduct never has nontrivial inverses. See this math.SE answer.