Up to this point I always just assumed a functor belonged to its domain category. It think this is because, categories are not closed under composition, and functors are basically "the way out" of the category.
But reading about diagrams I come across statements like "a diagram $F$ of type $C$ in $D$", where $F: C \to D$. So in the description of the diagram, it says "in $D$", so it seems $F \in D$.
I know that this is partly because of the viewpoint of having objects and morphisms of $C$ portrayed in $D$, but still.
Reading it up on Wikipedia, I noticed that they never mention it, either.