Still, what's in your opinion, the advantage of thinking in category theory rather than set theory? (For programming, not - algebraic geometry.)
I mean, all examples I heard can be directly treated with groups, monoids, and regular functions.
I know some abstract concepts that can be defined in a nice way with CT but not nearly as easy - set theory, e.g. (abstract) tensor product. Yet, for other concepts, including quantum mechanics, I have found that there is "abstract overhead" of CT with little added value.
In my opinion, the important advantage of category theory over set theory in (some!) computational contexts is that it allows you to generalize more easily. Generalizing from sets and functions to objects and morphisms lets you play around with instantiating those objects and morphisms with a variety of different beasts while maintaining the structure you've built up on top of them, and you can even use it to modularly build towers of functionality by layering one abstraction on top of another, even if you choose later to instantiate one of those layers with good old sets and functions. By contrast, it's hard to imagine treating something like async functions with plain old set theory: while there is of course a way to do it, you'd have to reason about several different layers of abstraction together to get all the way down to sets in one step.