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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.

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.