= Solution
Write $H_A=\mathcal C(-,A)$ for a <representable presheaf>. The contravariant <Yoneda lemma> states that, for every <presheaf on a category> $X$, evaluation at the identity is a bijection
$$
\boxed{\operatorname{Nat}(H_A,X)\longrightarrow X(A),\qquad\alpha\longmapsto\alpha_A(1_A),}
$$
natural in both $A$ and $X$. Smallness of $\mathcal C$ ensures the relevant families and natural-transformation collections are sets.
For $x\in X(A)$ define $\alpha^x_B(h)=X(h)(x)$ for each $h:B\to A$. This family is a <natural transformation>: for $u:B'\to B$, $X(u)\alpha^x_B(h)=X(u)X(h)x=X(hu)x=\alpha^x_{B'}(hu)$. Its identity value is $x$. Conversely, naturality of any $\alpha$ at $h:B\to A$ gives $\alpha_B(h)=X(h)\alpha_A(1_A)$. Thus evaluation and the displayed construction are inverse.
For a map $v:A\to A'$, precomposing a transformation $H_{A'}\to X$ with postcomposition $H_A\to H_{A'}$ corresponds to applying $X(v):X(A')\to X(A)$. For a transformation $\tau:X\to Y$, postcomposition corresponds to $x\mapsto\tau_A(x)$. These identities prove the stated <naturality> in both variables.
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