Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 17 / 3 / i / 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 $X=h_B$: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. The Yoneda embedding is full and faithful.