Left adjoint to a covariant representable functor
= Left adjoint to a covariant representable functor
{title2=$F(S)=\coprod_{s\in S}R$}
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.