Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note:

We will not distinguish between small and large categories. More generally, we will ignore any set-theoretic difficulties. These can be overcome with standard arguments using universes.

Question : Which of the notions introduced in Angelo Vistoli's notes assumes that the category is small? In particular their application to Algebraic/differentiable/topological stacks?

For example, Behrang Noohi puts the following extra condition in his notes on topological stacks:

Throughout the paper, all topological spaces are assumed to be compactly generated.

This could be because, the category $\text{Top}$ of all topological spaces is not a small category.

Are there any places one has to be careful to not allow large categories?