I do functional analysis, and diagrams are popping all over the place. It is about time I learned me some category theory.
As requested, making comment into the answer. I think Helemskii's book "Lectures and Exercises on Functional Analysis" contains a very nice intro into category theory. It may be a bit light on the algebraic side, but it certainly does contain a lot of analytic examples and motivations, and once you get the ball rolling and have some favourite examples in mind you can study any classical text on category theory at your leisure.
You could also enjoy reading his book "Quantum Functional Analysis: Non-Coordinate Approach". It is certainly about analysis rather than category theory, but the guiding principles of CT are always kept in mind and some relevant category-theoretic questions are considered.
I could suggest Tom Leinster's book: "Basic Category Theory", Cambridge studies in advanced mathematics 143. It is a modern introduction to category theory which covers the basic topics of the subject (I had to write an abstract for the Mathematical Reviews so I got it for free). It is not specifically written for people doing functional analysis. It is written for undergraduate students. It contains a lot of examples.