Hi Tim,
Jeff did not mention the excellent very categorical lectures given on operator spaces by Matthias Neufang at the FIELDS INSTITUTE Summer School in Operator Algebras held last summer at the University of Ottawa and that we both attended.
There are, of course, many people I could have credited and cited in my previous posting, but only the newest readers of this list will be unaware of the perils with which such an attempt is fraught. [For instance, I first read of the local presentability of Ban in Adamek and Rosicky's book, but I would not like to hazard a guess as to the origin of this result.] But you are right: I should have made an exception in Matthias' case. I should also credit Vladimir Pestov, a topologist who knows enough category theory to wonder whether operator spaces might be internal Banach spaces in some Grothendieck topos (but perhaps not enough to realise that this might take a student more than one term to prove), for having introduced me to Matthias several years ago. While at Dalhousie, I gave a talk on Pestov's conjecture; but when I started writing up my notes, I was distracted by an unrelated observation about the category of operator spaces which ultimately led to my ill-fated C*-algebra paper. I still haven't gotten back to the original project.
I do not know if any version of Matthias' notes is available.
Nor do I, but I am sure he would rather point people towards the pre-Wikipedia-era "online dictionary" of operator space theory to which he contributed: [German] http://www.math.uni-sb.de/ag/wittstock/projekt99.html [English] http://www.math.uni-sb.de/ag/wittstock/projekt2001.html These notes are quite good in the sense that, to use Tim's words, they are
explicitly conscious of the categorical content
In particular, it is quite gratifying to see a theorem such as "the forgetful functor from operator spaces to Banach space admits both a left and a right adjoint" stated (more or less) ungrudgingly. Cheers, Jeff. P.S. I should say that I don't think that operator spaces would be a suitable topic for an introductory CT course to a general audience; they are rather intricate. But it might be possible to craft an interesting set of exercises for the functional analysis contingent of such a course around operator space theory.