The first thing that got me really enthusiastic in category theory was seeing the definition of the categorical product in terms of universal properties. It's a very simple, but very different way of defining objects that makes it easy to see the relationships between similar objects in different contexts. For example, multiplication, the cartesian product, least common multiple, logical conjunction (&&), and structs (or record types) in programming are all products in particular categories* and the universal property definition unifies them very nicely. There isn't really enough space here to spell it out in detail, but this[0] is a good explanation. There is also a very natural way to manipulate the definition (categorical duality) to get coproducts which unify things like logical disjunction (||), greatest common divisor, and disjoint union of sets. This extremely unified view is also nice because if you have an unfamiliar mathematical or computational object, but you know that it has a categorical product, you suddenly know a ton about what you can do with it as well as some interesting questions to ask and properties to go looking for.

This genre of abstraction is all over the place in category theory and it gets way more interesting, but seeing products defined like this and how incredibly unifying of an abstraction it is was the first place that I really saw what the categorical perspective brought to the table.

*Cartesian product within the category of sets and functions between them (amounting to multiplication of cardinals, if one just cares about the action on objects), least common multiple within the category of positive integers ordered by divisibility (a partial ordering being just a special kind of category), logical conjunction within the category of truth values (which can be thought of as sets with at most one element), and structs or record types in the category whose objects are the types of your favorite programming language and morphisms are the programs between them. (From Chinjut, last time I brought up categorical products on hn: https://news.ycombinator.com/item?id=8780786)

[0] https://rnhmjoj.github.io/category-theory-for-programmers/sr...

If you'd to see how this applies more directly to programming, work like this is a direct application of the same definitional strategy for abstraction: https://blog.sumtypeofway.com/posts/introduction-to-recursio...