I have a notion from the real world I'm having trouble specifying categorically.
I have a functor $Free(G) \overset{F} {\longrightarrow} D$, where $Free(G)$ is the free category on some directed multi-graph $G$ (so it has a set of generating morphisms).
I'm interested in a sort of a functor-like mapping $S: D \rightarrow \mathbf{Set}$. I know how it acts on objets. The thing is, on morphisms, I want to specify action of $S$ on some set of generating morphisms and have the rest be defined as compositions of those. But I don't know anything about category $D$! I don't know any relations between morphisms in $D$ - so it seems $S$ can't be a functor defined in this way.
However, I do know that I'll always be able to define $S$ on the image of $F$, whatever it is, since there's a notion of a set of generators $G$ to the morphisms in this image.
How do I go about defining $S$ then? Can I even? I have a strong hunch it is a functor ,but I'm not sure how to go about defining it. Should it have $D$ as the domain?
It seems a functor factorization system such as this one, might be of use. Perhaps I should be concerned with defining a functor with a different domain $E \rightarrow \mathbf{Set}$?