Solution
= Solution
The <Yoneda embedding> sends $A$ to $h_A=\mathcal C(-,A)$ and $u:A\to B$ to postcomposition $h_u:h_A\to h_B$. Apply the <Yoneda lemma> with $X=h_B$:
$$
\operatorname{Nat}(h_A,h_B)\cong h_B(A)=\mathcal C(A,B).
$$
The inverse sends $u$ precisely to $h_u$, because its component at $C$ maps $f:C\to A$ to $uf$. Hence the induced maps on all <hom-sets> are <bijections>. \b[The Yoneda embedding is full and faithful.]