What seems the advantage to me is that category theory lets you account for more stuff within the formalisms.
Case in point, I did my dissertation in graph theory, and what I can tell you from proving hundreds of statements about graphs is that most of the theory of how and why graphs behave certain ways exists outside what can be captured formally by the theory. This is often what frustrates people about graph theory and all of combinatorics: the proofs come generally not by piling lots of previous results but by seeing through to some fundamental truth about what is going on with the collection (“category”) of graphs you are interested in and then using one of a handful of proof tricks (“functors”) to transform one thing into another to get your proof.
What seems the advantage to me is that category theory lets you account for more stuff within the formalisms.
Case in point, I did my dissertation in graph theory, and what I can tell you from proving hundreds of statements about graphs is that most of the theory of how and why graphs behave certain ways exists outside what can be captured formally by the theory. This is often what frustrates people about graph theory and all of combinatorics: the proofs come generally not by piling lots of previous results but by seeing through to some fundamental truth about what is going on with the collection (“category”) of graphs you are interested in and then using one of a handful of proof tricks (“functors”) to transform one thing into another to get your proof.