= Solution
For an <adjunction> $L:\mathcal C\rightleftarrows\mathcal D:R$, with unit $\eta$ and counit $\varepsilon$, call the adjunction idempotent when the multiplication
$$
\mu=R\varepsilon L:(RL)^2\longrightarrow RL
$$
of its induced <monad> is invertible. We show that this <idempotent adjunction> condition is equivalent to invertibility of $\delta=L\eta R:LR\to(LR)^2$, the comultiplication of the induced <comonad>.
For an <idempotent monad> $T$, the unit laws imply
$$
T\eta_A=\eta_{TA}=\mu_A^{-1}.
$$
If $a:TA\to A$ is a <monad algebra>, naturality of the unit and its algebra unit law give
$$
\eta_Aa=Ta\,\eta_{TA}=Ta\,T\eta_A=T(a\eta_A)=1_{TA},\qquad a\eta_A=1_A.
$$
Thus $\eta_A$ is an isomorphism. Apply this to the canonical algebra $R\varepsilon_B:T(RB)\to RB$: its unit law is a <triangle identity for an adjunction>. It follows that $\eta_{RB}$, and therefore $L\eta_{RB}$, is invertible for every $B$. Hence $\delta$ is invertible.
Apply the same argument to the dual adjunction $R^{\mathrm{op}}\dashv L^{\mathrm{op}}$. It gives the converse. \b[Idempotence is self-dual: the induced monad is idempotent exactly when the induced comonad is idempotent.] In particular, idempotence requires the unit to be invertible on every object in the image of the right adjoint.
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