Let $F:C\to D$ be a fully faithful functor, so that for each pair of objects $c_1,c_2\in C$ we have a bijection:
Now let $G_1,G_2:B\to C$ be any functors and suppose that we have a natural isomorphism of functors $F\circ G_1\cong F\circ G_2$. I understand that this induces isomorphisms $G_1(b)\cong G_2(b)$ in $C$, one for each object $b\in B$, but are these isomorphisms natural in $b$? That is, do we have a natural isomorphism of functors $G_1\cong G_2$?
If not, what additional hypothesis on $F$ is necessary? Is there a standard name for this concept?