Another reason to care about Category Theory, from a comment by Aloizio Macedo in an online math discussion:

What point am I trying to make? The point is: the important property that categories are really abstracting is structure itself. Category theory endows a mathematician with a capability to soar through the fields of mathematics with a broad vision. More than "inheritance of properties", there is identification of a common underlying structure. The naturality of such identification comes with time and experience.

And all the above is an understatement. The distillation process which category theory provides is a powerful tool in several aspects. For example, it is common to have a situation where we want things to exist and they are imposed by some condition. And the imposition commands the definitions, and proofs are matter of routine checking. ...

(cf Greatest Inventions (2011-06-09), Cakes, Custard, and Category Theory (2016-02-14), Category Theory Concepts (2016-04-25), Bird's-Eye View (2016-07-20), Category Theory for Programmers (2017-05-12), Ultimate Abstraction (2017-08-24), Put the Vast Storehouse in Order (2017-10-04), Yoneda Perspective (2018-10-03), Category Theory is like a Lighthouse (2018-12-24), Macro vs Micro (2019-02-03), ...) - ^z - 2019-03-22