= Idempotent adjunction
{title2=$R\varepsilon L:(RL)^2\cong RL$}
An <adjunction> $L\dashv R$ 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 $\eta_Aa=Ta\,\eta_{TA}=T(a\eta_A)=1$. Thus every algebra unit is invertible. Apply this to $R\varepsilon_B$ to obtain the self-dual adjunction condition.
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