To properly talk about compositionality in differential equations you absolutely need category theory. Likewise I firmly believe that PDE and numerical mathematics actually lack a good foundation in category theory, not that they don’t need one.

It is just very hard to create non-trivial general theories in sich „Applied“ fields, Functional Analysis alone is not very exciting. For an example see for example the work or Hairer (stochastic differential equations) or Costello (QFT).