What do we call the domain and codomain of a functor? Do they have special names? Do the domain and codomain categories of a functor have special names, or are they simply called the domain and codomain? It seems to me that it might not be appropriate to call them as such, since we usually do this when we're talking about functions, whose domains have to be sets. But the "domain" of a functor need not be a set.

The usual terminology for morphisms (so in particular functors) in category theory is that of source and target. So if $C,D$ are categories and $F\colon C\rightarrow D$ is a functor, you can refer to $C$ as the source (or source category, if there is possibility of ambiguity) of $F$ and $D$ as the target (or target category) of $F$.
