Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (linklink). I'm wondering if there are any results in AG that actually depend on the existence of universes (and what some of the more interesting ones are).
I'm of course aware of the result that as long as we require that the classes of objects and arrows are sets (this is the only valid approach from Bourbaki's perspective), for every category C, there exists a universe U such that the U-Yoneda lemma holds for U-Psh(C) (this relative approach makes proper classes pointless because every universe allows us to model a higher level of "largeness"), but this is really the only striking application of universes that I know of (and the only result I'm aware of where it's clear that they are necessary for the result).
