= Solution
The <Yoneda embedding> is the <functor> $H_\bullet:\mathcal C\to[\mathcal C^{\mathrm{op}},\mathbf{Set}]$ sending $A$ to $H_A=\mathcal C(-,A)$ and $f:A\to B$ to postcomposition, $(H_f)_U(h)=fh$. Composition and identities are preserved by associativity and the identity laws in $\mathcal C$.
Apply the <Yoneda lemma> with $X=H_B$. It gives a bijection $\operatorname{Nat}(H_A,H_B)\cong\mathcal C(A,B)$. The inverse sends $f$ precisely to $H_f$, since $H_B(h)(f)=fh$. Hence every transformation between the representables comes from exactly one morphism of $\mathcal C$. This proves that \b[$H_\bullet$ is a full and faithful <functor>].
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