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**Math Major** Let $\displaystyle C$ be a category with $\displaystyle Obj(C) = \mathbb{Z}^+ $ and for two positive integers $\displaystyle a $ and $\displaystyle b$, there is exactly one morphism $\displaystyle a \rightarrow b $ iff $\displaystyle a$ divides $\displaystyle b$ without remainder, and $\displaystyle Hom(a,b) = \emptyset$ 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.