Well, not exactly. Things till functions and functor are part of almost every big enough theory like set theory or even just natural numbers and used everywhere

Well, not exactly. Things till functions and functor are part of almost every big enough theory like set theory or even just natural numbers and used everywhere, and if arrows are only used for describing that, I wouldn't say it is application of category theory. Only point of category theory starts when you go higher than that and deal with more abstract and generalizable things e.g monads(which again is used in other theories), bigger infinities, ordering of infinities etc.