ct.category theory - conditions for natural transformations to exist?

Given a category $C$ with two objects and one non-identity morphism

$$ato b$$

and another similar category $D$

$$xto y,$$

we can define two functors $F,G:Cto D$ with

$$F:amapsto x, bmapsto y$$

and

$$G:amapsto x, bmapsto x$$

with morphisms doing the only thing they possibly can.

A natural transformation $alpha:FRightarrow G$ would require a component $alpha_b:F(b)to G(b)$, but there is no morphism $yto x$, 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?