I know that the same question was asked before, but I'm looking for some hints to get started on the question, rather than a full solution.
The question is given as
My attempt:
I'm not sure how to begin this, but my attempt for ii) at least has been to note that since $F$ is naturally isomorphic to $C(c, -)$(the functor that sends every element of $c \in C$ to the Hom set $C(c,x)$) for some $c \in C$, $GH$ is also naturally isomorphic to $C(c,-)$, since natural isomorphism is transitive.
This implies that $GH$ is representable, which would seem to imply that $G$ is also representable, which I have not proved yet.
As for i), I'm not where to start, but my intuition is that it's false. But coming up with counter-example would be whole another issue.
Hints would be appreciated a lot, thanks!
Here's a hint for part ii). You have an object $c$ that represents $F$. To show that $G$ is representable, we should look for an object to represent it. Well, seeing as we have $c\in C$ and a functor $H:C \to D$, we have an object $H(c)\in D$. Perhaps this will represent $G$?
For part i), it might be useful to choose an inverse to $H$.