Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 4 / c / Solution

By part (b), $i:P\to H_B$ is the equaliser of $ip$ and $1_{H_B}$. Full faithfulness of the Yoneda embedding writes $ip=H_b$ for a unique endomorphism $b:B\to B$. Choose an equaliser $e:A\to B$ of $b$ and $1_B$ in $\mathcal{C}$.

For each $c$, the maps $c\to A$ are precisely the maps $u:c\to B$ with $bu=u$. Thus $H_e:H_A\to H_B$ is the pointwise equaliser of $H_b$ and $1_{H_B}$, hence their equaliser in the presheaf category. Both $i$ and $H_e$ have the same universal property, so their vertices are isomorphic. This proves the representable retract criterion: