I want these functors to have the following properties, they seem a bit arbitrary though - so I was looking for sufficient "standard" properties of functors which imply them (such as full, faithful etc.)
Let $F:C \to D$ be a functor with the following property. If $F(c)=F(c')$ then there exists a $f: c \to c'$ in $C$ such that $F(f)=\text{id}_{F(c)}$.
Let $G:C \to D$ be a functor with the following property. For all $f:c \to c'$ in $C$, if $G(f)=\text{id}_{F(c)}$ then $f= \text{id}_{c}$.
Thanks!