= Solution
Suppose $U(C)\cong\mathcal C(R,C)$ naturally and $\mathcal C$ has small <coproducts in a category>. Define
$$
F(X)=\coprod_{x\in X}R.
$$
For a function $v:X\to Y$, define $F(v)$ by sending the $x$-summand by the identity of $R$ to the $v(x)$-summand. The <coproduct> property gives the identity and composition laws, so $F$ is a <functor>. There are natural <bijections>
$$
\mathcal C(F(X),C)\cong\prod_{x\in X}\mathcal C(R,C)
\cong\mathbf{Set}(X,U(C)).
$$
The first chooses a <morphism> on every summand; the second views that family as a function. Hence this is the <left adjoint to a covariant representable functor>:
$$
\boxed{F\dashv U,\qquad F(X)=\coprod_{x\in X}R.}
$$
For empty $X$, the formula uses the empty <coproduct>, an <initial object>, and the same bijection remains valid.
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