Of course, the chain rule explained via differential topology is simply functoriality!

IME, category theory was much more valuable in differential topology in helping solidify why the structures and their compositional properties make sense. You get to work with several, honest-to-god categories and functors between them, especially when tangent bundles come into the mix.

Now go to programming, there's really only one category at play, and I'm not sure how worthwhile it is to apply a (more contrived) version of CT that only has endofunctors and being cartesian closed.