Left adjoint to a covariant representable functor ($F(S)={\coprod }_{s\in S}R$)

If $\mathcal{C}$ has small coproducts in a category and $U\cong \mathcal{C}(R,-)$, then $F(S)={\coprod }_{s\in S}R$ is a left adjoint to $U$. Maps from this coproduct to $X$ are families of maps $R\to X$, naturally equivalent to functions $S\to UX$. Conversely, a set-valued right adjoint $U$ is represented by $F(1)$, because maps from a singleton evaluate to elements.