Let $R$ be a commutative ring. Denote by $R$-Mod the category of $R$ modules.
$R$ is described by a universal property, it is the free $R$-module on one generator. This universal property is captured by the fact that $\text{Hom}_R(R,{-})$ is naturally isomorphic to the obvious forgetful functor $U : R\text{-Mod} \to \text{Sets}$.
Is the functor $\text{Hom}_R(-,R)$ naturally isomorphic to some well known functor. If so what object does it represent?