One interesting thing is that category theory provides a way to precisely describe a Most General Unifier. It is a example of a coequalizer.

This isn't a really basic and accessible result, but IMO it gives a good flavour of what category theory is suitable for: formally describing constructions we didn't previously have the tools to express precisely.

https://www.sciencedirect.com/science/article/abs/pii/B97801...