First a disclaimer - I am NOT an analyst, and MUCH of this stuff goes way over my head. Most what interests me about this stuff is when I can use categories help to /substitute/ other types of intuition for analytical intution.
That aside, here are some potentially useful references:
1. Helemski's functional analysis book is heavily categorical:
https://b-ok.cc/md5/CAED8B841C25404ACE856757F5C10D15
2. In general banach spaces have some interesting categorical background, see or example
i) Castillo's Hitchiker's Guide to Categorical banach space theory https://www.eweb.unex.es/eweb/extracta/Vol-25-2/25J2Castillo.pdf
ii) Tom Leistner's note:
https://www.maths.ed.ac.uk/~tl/glasgowpssl/banach.pdf
iii) https://arxiv.org/pdf/1302.3372.pdf
3. Gustave Chouquet's remarkable 3 book series Lectures on Analysis starts with an intro to categories and limits/colimits (traditionally called inductive/projective limits in analysis and topology), and in general is good about pointing out when the objects of analysis form a category
i) book 1: http://library1.org/_ads/11F3A5549B8073C17481393598026F5B
ii) book 2:
http://library1.org/_ads/0C73C7A22B66C7DFBB03D800FD84EB59
iii) book 3: http://library1.org/_ads/5707F1BEFD5AC89F0D758549E7C620D3
4. The theory of coalgebras and bisimulationss provides an important unification of various models of dynamical systems processes. Dynamical systems, markov chains, markov decision processes, stochastic processes, quantum processes, and concurrency theories - and more - are all coalgebras or comonads of various functors. A categorical view lets us, for instance, optimize efforts "transfer" processes from one setting to another.I'm away from my bibliography on this topic, but see for example:
i) https://sci-hub.tw/10.1016/j.entcs.2009.02.012
ii) https://link.springer.com/content/pdf/10.1007/978-0-387-35520-7_20.pdf
iii) http://www.cs.ox.ac.uk/people/samuel.staton/papers/calco09.pdf
5. Categories come up a bit in organizing Harmonic analysis
i) Folland course in abstract harmonic analysis uses them JUST a bit to point out that harmonic analysis works with a functor from abelian groups to the compact abelian groups (admittedly not a great example!)
http://alirejali.ir/afiles/up/other/book5/A%20Course%20in%20Abstract%20Harmonic%20Analysis.pdf
ii) This workshop (with video lectures!) talks about the Laglands program and its applications to Harmonic analysis
http://www.msri.org/workshops/708