= Solution
Define the <Yoneda embedding> $H_\bullet:\mathcal C\to[\mathcal C^{\mathrm{op}},\mathbf{Set}]$ by $H_A(U)=\mathcal C(U,A)$ and $H_A(k)(h)=hk$. For $f:A\to B$, define $H_\bullet(f)_U(h)=fh$. Associativity proves <naturality> in $U$, and preservation of <identity morphisms> and <composition in a category> proves that $H_\bullet$ is a <functor>.
Apply the <Yoneda lemma> to $X=H_B$. It gives
$$
\boxed{\operatorname{Nat}(H_A,H_B)\cong\mathcal C(A,B).}
$$
The <natural transformation> corresponding to $f$ has component $h\mapsto fh$, exactly $H_\bullet(f)$. Therefore the induced map on every <hom-set> is a <bijection>: \b[the Yoneda embedding is <full and faithful>].
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