I'm obliged, I suppose, to list off a few big wins. From the 50s and 60s, we have classic theorems which use abstract nonsense to generalize big statements about entire classes of objects, like Freyd's adjoint functor theorem [0], Yoneda's lemma [1], and Lawvere's fixed-point theorem [2]. Yoneda's lemma is the slogan that "an object is equivalent to the arrows coming/leaving it", but formal. Lawvere's theorem is a deep and permanent generalization of Gödel, Tarski, Turing, et al. on incompleteness and undecideability.
Starting in the 60s and continuing to the present, there's been a theme of exploring category theory as a prime foundation for maths. There's been complete foundations like the Elementary Theory of the Category of Categories (ETCC) [3], and also much smaller but pointed presentations that focus on just simplifying set theory. I like [4] in particular.
On a deeper level, in terms of structure and philosophy, the entire existence of homotopy type theory (HoTT) relies on categorical presentations and memes. HoTT is, more than any other type theory, an investigation into what equivalence means, and it dovetails wonderfully with 2-category theory and the understanding that algebraic laws can be transformed into transformations.
[0] https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_fu...
[1] https://en.wikipedia.org/wiki/Yoneda_lemma
[2] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf
[3] https://ncatlab.org/nlab/show/ETCC
[4] https://arxiv.org/abs/1212.6543
Starting in the 60s and continuing to the present, there's been a theme of exploring category theory as a prime foundation for maths. There's been complete foundations like the Elementary Theory of the Category of Categories (ETCC) [3], and also much smaller but pointed presentations that focus on just simplifying set theory. I like [4] in particular.
On a deeper level, in terms of structure and philosophy, the entire existence of homotopy type theory (HoTT) relies on categorical presentations and memes. HoTT is, more than any other type theory, an investigation into what equivalence means, and it dovetails wonderfully with 2-category theory and the understanding that algebraic laws can be transformed into transformations.
[0] https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_fu...
[1] https://en.wikipedia.org/wiki/Yoneda_lemma
[2] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf
[3] https://ncatlab.org/nlab/show/ETCC
[4] https://arxiv.org/abs/1212.6543