The Yoneda lemma states that for a locally small category $C$ and a functor $F: C \rightarrow Set$, for each object $c \in C$, there is a bijection between the set of natural transformations $[C, Set](Hom(c, -), F)$ and the set $F(c)$.
In particular, this talks about natural transformations from $Hom(c, -)$ to $F$. Can we talk about the reverse? I am looking for a theorem of the form:
For a locally small category $C$ and a functor $F: C \rightarrow Set$, for each object $c \in C$ such that
???, there is a bijection between the set of natural transformations $[C, Set](F, Hom(c, -))$ and the set $F(c)$.
If such a result is not possible, I'd like to understand what the obstacle to having such a result is.