= Solution
Choose a representing object $R$ and a <natural isomorphism> $\theta_X:\mathcal C(R,X)\cong U X$. Using the assumed small <coproducts in a category>, define
$$
\boxed{F(S)=\coprod_{s\in S}R.}
$$
For a function $v:S\to S'$, define $F(v)$ by $F(v)\iota_s=\iota_{v(s)}$. The <coproduct in a category> uniqueness clause proves preservation of identities and composition, so this is a <functor>.
Restriction to the coproduct summands, followed by $\theta$, gives
$$
\mathcal C(F(S),X)\cong\prod_{s\in S}\mathcal C(R,X)\cong\mathbf{Set}(S,U X).
$$
These <bijections> are natural in $S$ by the definition of $F(v)$, and natural in $X$ by naturality of $\theta$. They establish \b[$F\dashv U$], the <left adjoint to a covariant representable functor>. The empty set is sent to the empty coproduct.
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