Category on integers with the usual product and coproduct?

Category on integers with the usual product and coproduct? References 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.