Consider a locally small category $C$ and the functor $Hom_C(A,-):C\rightarrow Set$. What can we say about its natural transformations?
My attempt: Yoneda's lemma tells me that given a functor $F:C\rightarrow Set$
We have $Nat(Hom_C(A,-),F)$ isomorphic to $F(A)$. In this case $F=Hom_C(A,-)$ so the natural transformations are simply the set of functions from $A$ to $X\in C$?
My reasoning looks inconsistent/incoherent that' why I am asking for help.
Indeed, by the Yoneda Lemma $\text{Nat}(\text{Hom}_{\mathcal{C}}(A,-),\text{Hom}_{\mathcal{C}}(A,-))\cong\text{Hom}_{\mathcal{C}}(A,A)$, thus the set of natural transformations between $\text{Hom}_{\mathcal{C}}(A,-)$ and $\text{Hom}_{\mathcal{C}}(A,-)$ is in natural bijective correspondence with $\text{Hom}_{\mathcal{C}}(A,A)$. The set $\text{Hom}_{\mathcal{C}}(A,A)$ is the set of all morphisms from $A$ to $A$ in $\mathcal{C}$ (they are called endomorphisms of $A$ in $\mathcal{C}$). Note, that $\text{Hom}_{\mathcal{C}}(A,A)$ is not (in general) a set of some "functions" (in any sense).