Similarly you can show that gcd and lcm are associative, once you know their universal properties (for gcd, being the one common divisor that every common divisor divides; for lcm, the corresponding dual property). Similarly the associativity (up to unique diagram-fitting isomorphism) of products (or even fibered products) in any category (when they exist).
The really high horse here is called "Yoneda embedding", but there is not much gained from going that high.
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.
The really high horse here is called "Yoneda embedding", but there is not much gained from going that high.