I want to show that for any natural transformation $\eta:\text{id}_\mathsf{Ring}\to\text{id}_\mathsf{Ring}$ we have that $\eta_R=\text{id}_R$ for all $R\in\text{ob}(\mathsf{Ring})$.
I was able to show that in $\mathsf{Grp}$ there are only two natural transformations $\text{id}_\mathsf{Grp}\to\text{id}_\mathsf{Grp}$ (the identity and the zero morphisms) by showing that group homomorphisms $\mathbb{Z}\to G$ are in bijection with elements of $G$ so $\eta_G(g)=\eta_\mathbb{Z}(1)g$ for all $G$, and this is a group homomorphism for non-abelian groups if and only if $\eta_\mathbb{Z}(1)=0,1$.
I tried to do something similar with rings, but $\mathbb{Z}$ doesn't work and I am not sure which ring to use instead. Is this the right approach or is there a better way? I'm looking for a hint rather than a full solution