 Solution

Suppose $F:\mathbf{Set}\to \mathcal{C}$ is a left adjoint to $U$. The adjunction gives bijections, natural in $X$,The second map is evaluation at the singleton element, as in part (a). Thus $U$ is represented by $F1$. This argument uses the one-point set as a generator of the particular set-valued adjunction; it does not claim that every arbitrary right adjoint is representable.