It was asked earlier (Why Et(X) is not small?) why $Et(X)$, the category of schemes etale over a fixed scheme $X$ is not small. I was wondering how does this issue come up in practice? I'm guessing somewhere in the development of say etale cohomology (which I don't know much about), one needs to work with a category that is small. What is an example of such an instance?
The following similar question had useful answers Importance of 'smallness' in a category, and functor categories
and one of the answers said that to form the category of presheaves on a category $C$ (such as $C=Et(X)$) you need $C$ to be small, since if $C$ is a proper class, then the presheaf category $[C^{op}, Sets]$ is certainly not small and someone suspected it might not even be a class. But in the development of etale cohomology, are there results where we really need to consider all presheaves on $Et(X)$ at once, and not just some (say a class or set) of presheaves?
Sheaves associated to presheaves usually only exist on small sites.