Let be a category with and for two positive integers and , there is exactly one morphism iff divides without remainder, and otherwise. In this category, verify that there are products and coproducts. What are they?

My question - I've worked through it, and I came up with the products being the greatest common divisor of two integers and the coproducts the least common multiple. I just wanted to make sure I had come up with the correct interpretation before moving on.