 Solution

Choose a representing object $R$ and a natural isomorphism ${\theta }_X:\mathcal{C}(R,X)\cong UX$. Using the assumed small coproducts in a category, defineFor a function $v:S\to S^{\prime }$, 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 $\theta$, givesThese bijections are natural in $S$ by the definition of $F(v)$, and natural in $X$ by naturality of $\theta$. They establish $F⊣U$, the left adjoint to a covariant representable functor. The empty set is sent to the empty coproduct.