From Mac Lane:
Let $K: D \rightarrow \text{Set}$ be a functor with $D$ having small hom sets, the bijection:
$y : \text{Nat}(D(r,-), K) \cong Kr$ by $\alpha : D(r,-) \rightarrow K$ to $\alpha_r 1_r$
is a natural isomorphism $y : N \rightarrow E$ between the functors $E, N : \text{Set}^{D} \times D \rightarrow \text{Set}$, where $E: \langle K, r \rangle \mapsto Kr$ and $N : \langle K, r \rangle \mapsto \text{Nat}(D(r,-) \rightarrow K)$
I'm trying to show that $y : N \rightarrow E$ is a natural isomorphism but I'm having trouble with one part.
I need to show that for any $\langle F, f \rangle : \langle K, r \rangle \rightarrow \langle K', r' \rangle$: $$E \langle F,f \rangle \circ y_{\langle K, r\rangle} = y_{\langle K', r' \rangle} \circ N \langle F, f \rangle$$ when evaluated at $\alpha : D(r, -) \rightarrow K$.
This should reduce to: $$E \langle F, f \rangle (\alpha_r 1_r) = \alpha ' _{r'} 1_{r'}$$ but I'm having trouble showing the equality between these two things.
Anyone have any ideas?