The analogy between functions from a set to set, and functors from a category to a category is obvious. But transferring terminology from functions to functors can be a tricky business.

I would like to ask if the following is acceptable: For $B$ a subcategory of $A$, and $D$ a subcategory of $C$, and $F:A \to C$ a functor, is the following statement clear in meaning, and well-defined?

It holds that the image of $B$ under $F$ lies in $D$.

If not, then how should it be phrased?

isa functor (between discrete categories). – Qiaochu Yuan Sep 19 '12 at 16:15