 Idempotent adjunction

An adjunction $L⊣R$ is idempotent when its induced monad has invertible multiplication. This is equivalent to its induced comonad having invertible comultiplication. For an idempotent monad, the two maps ${\eta }_{TA}$ and $T{\eta }_A$ agree, since both invert the multiplication. If $a:TA\to A$ is an algebra, naturality then gives ${\eta }_{A}a=Ta{\eta }_{TA}=T(a{\eta }_A)=1$. Thus every algebra unit is invertible. Apply this to $R{\epsilon }_B$ to obtain the self-dual adjunction condition.