Given a category $C$ with two objects and one non-identity morphism
and another similar category $D$
we can define two functors $F,G:C\to D$ with
$$F:a\mapsto x, b\mapsto y$$
$$G:a\mapsto x, b\mapsto x$$
with morphisms doing the only thing they possibly can.
A natural transformation $\alpha:F\Rightarrow G$ would require a component $\alpha_b:F(b)\to G(b)$, but there is no morphism $y\to x$, so if I understand this correctly, there is no natural transformation from $F$ to $G$.
Is that correct? Is there a clear set of criteria required for there to exist a natural transformation?