I'm having difficulty understanding the Yoneda lemma. In particular, the proof isn't that obvious to me. Please, could someone explain to me the error of my current understanding..
The Yoneda lemma says that if $F$ is a functor from $C \rightarrow Set$, then $F a$ is isomorphic to the natural transformations from $hom(-,a)$ to $F$. Now, the part that gets me is if I say $F$ is a functor which sends objects of $C$ to the same single-element set and morphisms to the identity function, how can this bijection exist. There is one element in $F a$, but multiple natural transformations (e.g. $C(-,a)$ to $F$, and $C(-,b)$ to $F$).
Please someone help me.