I should have demonstrated my picks. I didn't just look for citation but also looked for at least some nontrivial review of the results of the paper^.
I pick these two papers. The first is literally about adjunctions pertaining to combinatorial species whereas the second devotes an entire section to reviewing the theory and stating results that it uses in a way that I don't really understand. I'm going to read the first paper though because it's relevant to my interests :)
Rajan. The adjoints to the derivative functor on species
The fact that I have to go rummaging around for these examples kind of proves your point, doesn't it? I don't think category theory will lead to any fantastic new results in the fields we're discussing, but the bar is quite low for it to be useful as an organizational tool.
OK. The first paper is indeed literally about species and in a good but not top journal. But it's not connected to the mainstream of combinatorics in any way, in the sense that if you aren't a priori interested in species, there's no reason to be interested in the paper.
The second paper is interesting because it's in an excellent journal and about a problem not obviously connected with species. So I agree that it counts as a good example for your case. I also agree with your conclusion – it's an exception that proves the rule, so to speak. Most probabilists have no need for category theory, and most AoP papers don't use categories. (I feel that may literally be the only one?)
I also agree that, to the extent it's useful, it's useful as an organizing principle and not "substantively." I suppose this explains why I feel it's overhyped: mathematicians care about solving problems, and the insights that solve the problems ultimately have to come from some problem-specific observations.
I pick these two papers. The first is literally about adjunctions pertaining to combinatorial species whereas the second devotes an entire section to reviewing the theory and stating results that it uses in a way that I don't really understand. I'm going to read the first paper though because it's relevant to my interests :)
Rajan. The adjoints to the derivative functor on species
https://www.sciencedirect.com/science/article/pii/0097316593...
Panagiotou, Konstantinos; Stufler, Benedikt; Weller, Kerstin. Scaling limits of random graphs from subcritical classes.
https://projecteuclid.org/euclid.aop/1474462098
The fact that I have to go rummaging around for these examples kind of proves your point, doesn't it? I don't think category theory will lead to any fantastic new results in the fields we're discussing, but the bar is quite low for it to be useful as an organizational tool.
^ I searched for the word functor of course! :)