TITLE: Is there a name for categories whose objects are sets?
QUESTION [5 upvotes]: A category $C$ is small if $\operatorname{Obj}(C)$ and $\operatorname{Hom}(C)$ are sets.
A category is locally small if for all objects $a,b$, the morphisms $\operatorname{Hom}(a,b)$ is a set.
Is there a name for categories whose objects are sets? For example $\mathrm{Set}, \mathrm{Grp}, \mathrm{Top}$.

REPLY [2 votes]: Category theory doesn't really care what its objects are per se. So it's not common for terminology in the field to distinguish categories based on what the objects are. You can ask about categories who behave somewhat like the category of sets (or a subcategory thereof), but that won't tell you what the objects truly are.