Let $F$ be the functor from commutative rings to abelian groups that takes a commutative ring $R$ to its group of units.
I want to show that the functor that takes each group to the group ring ${\bf Z} G$ is left-adjoint to $F$ and am not sure whether this functor has a right adjoint. I think it does not.
Left adjoint functors preserve initial objects. If the functor has a right adjoint, it itself is a left adjoint. But the group of units of the initial object ${\bf Z}$ is $({\pm 1}, \cdot) \cong {\bf Z}/(2)$, a contradiction.