A functor F:C→D from a category C to a category D is said to be full if, for each pair of objects, x,y ∈ C, the function,
F:C(x,y)→D(F(x),F(y))
between hom sets is surjective.
What I am unsure about is, for categories C and D, can you have a functor that is not full in both directions? That is, a category F:C→D and G:D→C, such that both F and G are not full.
In particular, are there any simple examples of this?
Simplest example: Let $C = D$ be the category with one object $x$ and two arrows $\text{id}_x$ and $f$ such that $f\circ f=\text{id}_x$. (This is the cyclic group of order $2$ viewed as a category.) Let $F=G\colon C\to C$ be the functor sending $x$ to $x$ and both arrows to $\text{id}_x$.