How to refer to image of homset under a (not full, not faithful) functor.

Question

Consider a functor $F: C \rightarrow Set$. If $F$ is not full and not faithful, how can I talk about the image of $F$ on morphisms (a subset of $Hom_{Set}(F(c), F(c'))$) in relation to $Hom_C(c, c')$? Using a mix of set-theoretic and category theoretic terms, I might describe this subset of $Hom_{Set}(F(c), F(c'))$ as $Hom(c, c') / \sim$ where two morphisms in $Hom_C(c, c')$ are equivalent if their images are the same under $F$. I'd like to describe this kind of equivalence relation using a category theoretic construction.