Do you know of a textbook at the graduate level about the common categories of functional analysis (such as the 2 categories of Banach spaces, the 2 categories of Banach algebras, the category of Banach modules, the category of C$^*$-algebras, the category of W$^*$-algebras) which is suitable for a student who has taken a basic course in category theory?
The ideal book covers the basic themes such as injective, projective, flat objects; limits & colimits; tensor products; injective envelopes; etc. of each category in the category theoretical context, with their context-specific equivalent definitions. It would be instructive if the book contains examples for the (existence and) non-existence results.
I can't blame if you think that I have way too many criteria for the ideal book, so I can't match up with a great book that would fill up the gap in my life and my bookshelf.
Some recommended books:
(1) "Completely bounded maps and operator algebras" by Paulsen. In particular, chapter 15 contains a nice chapter about Hamana's theory of injective envelopes of operator spaces, a theory that has attracted much attention the past few years. I suggest complementing this chapter with Hamana's original papers, which are quite readable.
(2) "Compact quantum groups and their representation category". This is the reference book for the study of $C^*$-categories that admit a monoidal structure. Roughly, such a category has Banach spaces as morphism spaces and one can use techniques from C*-algebras to study these categories. Important (but not all) examples of such categories arise as the representation category of the so called compact quantum groups (which generalise compact topological groups), which explains the title of the book. This book is not easy though, but very rewarding once you get through it.
(3) "Amenable Banach algebras" by Runde. Maybe not exactly what you have in mind, but probably worth skimming through. For example, it has a chapter on homological algebra for Banach spaces.