I'd say Grothendieck's proofs of the Weil Conjectures is a good example. His proof uses etale cohomology and the definition of etale cohomoly uses Category Theo

I'd say Grothendieck's proofs of the Weil Conjectures is a good example. His proof uses etale cohomology and the definition of etale cohomoly uses Category Theory in a fundamental way. From the etale cohomoly wikipedia page https://en.wikipedia.org/wiki/%C3%89tale_cohomology https://en.wikipedia.org/wiki/%C3%89tale_cohomology "For any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X. It is an analogue of the category of open subsets of a topological space, and its objects can be thought of informally as "étale open subsets" of X. The intersection of two open sets of a topological space corresponds to the pullback of two étale maps to X. There is a rather minor set-theoretical problem here, since Et(X) is a "large" category: its objects do not form a set." There's a lot of advanced math in that paragraph, but it should be clear that Category Theory is needed to define etale cohomology.