This is a very naive question. Suppose I'm given two functors $F,G\colon(Schemes/S)^{op}\to (Sets)$, over some base scheme $S$, represented by $S$-Schemes $X_F$ and $X_G$ respectively. There are conditions for a natural transformation $\eta\colon F\to G$ to be representable by a morphism of the representing schemes, I know. But if there is an arbitrary natural equivalence $F\cong G$, then $h_{X_F}\cong F\cong G\cong h_{X_G}$ at least, which looks unlikely if $X_F$ and $X_G$ weren't isomorphic.
Is there any known natural equivalence between functors which are represented by non-isomorphic schemes? Or perhaps even more interesting, is there a non-representable natural equivalence between functors, represented by isomorphic schemes?
If there is an illustrative example in any other category that is not too weirdly constructed, it would be nice to see that too.
The Yoneda lemma tells us that if $F$ and $G$ are represented by schemes $X_F$ and $X_G$, then every natural transformation from $F$ to $G$ arises from a morphism of the representing schemes. In particular, every natural equivalence $F\cong G$ arises from an isomorphism $X_F\cong X_G$.