Others can definitely give better opinions, but I currently have "Lectures and exercises on functional analysis" checked out from the library, and I have been enjoying the few parts that I've read so far.
I can not comment on the use of category theory in analysis, but for people who aren't very comfortable with more abstract fields where category theory plays a major role a book like the one above is great since it goes over a lot of basic category theory while keeping the main characters from analysis. At the very least it's a great way to get accustomed to the language.
