Yoneda lemma, bijection between sets of natural transformations

Question

Let $Y: \cal A\to Set^{\cal A^{op}}$ be the Yonneda embedding and $S:Set^{\cal A^{op}}\to Set$ an arbitrary functor. How do I use the Yonneda lemma to obtain a bijection between natural transformations $hom(A,-)\to SY$ AND these: $hom(hom(-,A),-)\to S$.
See the second displayed formula Here.

Answer 1

Use the Yoneda lemma once to obtain $\text{Nat}(\hom(\hom(-,A),-),-),S) \cong S(\hom(-,A))$. Use it a second time to obtain $\text{Nat}(\hom(A,-),SY) \cong SY(A)$ and then notice that $SY(A)=S(\hom(-,A))$.