Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 17 / 7 / ii / Solution

Suppose $U(C)\cong \mathcal{C}(R,C)$ naturally and $\mathcal{C}$ has small coproducts in a category. DefineFor 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 bijectionsThe 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:For empty $X$, the formula uses the empty coproduct, an initial object, and the same bijection remains valid.