I am trying to prove that the forgetful (covariant) functor $U:\mathbf{Ring}\to\mathbf{Set}$, sending a given ring to its underlying set is representable. I know this functor is represented by the polynomial ring $\mathbb{Z}[X]$. Hence one needs to establish a natural isomorphism $$\eta:U\to \mathbf{Hom}_{\mathbf{Ring}}(\mathbb{Z}[X],-)$$ My difficulty is finding what the components of this natural transformation are, explicitly; in other words, where (to which ring homomorphism $\mathbb{Z}[X]\to R$) does $\eta_R$ send an element of (the set) $UR$.
Thanks for any help!