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

The Yoneda embedding sends $c$ to the representable presheaf $H_c=\mathcal{C}(-,c)$. On a morphism $f:c\to d$, it sends a map $u:a\to c$ to $fu:a\to d$, giving $H_f:H_c\to H_d$.

The Yoneda lemma states that for every categorical presheaf $P$ there is a bijection, natural in both variables,Its inverse sends $x\in P(c)$ to the transformation $u:a\to c\mapsto P(u)x$. Apply it to $P=H_d$ to obtainUnder this bijection $H_f$ corresponds exactly to $f$, so the map induced by the embedding on each hom-set is bijective. Therefore the Yoneda embedding is full and faithful.