Suppose $F : C \to Set$ is equivalent to $G: D \to Set$ in the sense that there is an equivalence of categories $H : C \to D$ so that $GH$ and $F$ are naturally isomorphic. If $F$ is representable, then $G$ is representable? If $G$ is representable, then is $F$ representable?
Is the answer to either of these yes? If so, why? If not, why not?
Write $K:\mathcal D\to\mathcal C$, for the "inverse" of $H$, i.e. $HK\cong Id_\mathcal D$ and $KH\cong Id_\mathcal C$.
Now we can note, unsurprisingly, that your questions are symmetric. If one holds, then both hold, because $G\cong GHK\cong FK$ follows from $GH\cong F$.
The question now is does $\mathcal D(D,H(-))\cong F$ imply that $F$ is representable. We have $$F\cong\mathcal D(D,H(-))\cong\mathcal D(HKD,H(-))\cong \mathcal C(KD,-)$$ The last is because $H$ is full and faithful.
If we rectify $H$ and $K$ to an adjoint equivalence, which we can always do, then we could go straight from $\mathcal D(D,H(-))\cong\mathcal C(KD,-)$ via the adjunction.