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!

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!