You don't really need to bring the Yoneda embedding into it. They're all just partial orders (for the gcd and lcm use the order defined by divisability), which

You don't really need to bring the Yoneda embedding into it. They're all just partial orders (for the gcd and lcm use the order defined by divisability), which can easily be shown to be categories with at most a single function between objects. After that it's just expanding definitions.