Hey there, Jesus! You seem to be doing quite well figuring out what I was hinting at. I was too lazy to explain all this stuff, but you found a page on the nLab where I was explaining some of it to Mike Stay once upon a time, so it's okay that I'm a bit less energetic now. If you have any questions, just ask!

And yes, the size issues are annoying here: it's really tempting to think of 'free cocompletion' as a monad on \\(\mathrm{Cat}\\), but in fact it maps small categories to large ones, and I guess large ones to extra-large ones (though I haven't even thought about that), and so on. Gambino, Hyland and Power were writing a very nice paper that stumbled into this tar pit, and instead of just saying 'pretend size issues aren't really a problem' (as I'd be tempted to do), they dove in and decided to generalize the whole theory of monads to deal with this problem. I'm glad to hear they finally did it!