Discrete-indiscrete adjunction ($D⊣J:\mathbf{Top}\to \mathbf{Top}$)

Discretization $D$ and indiscretization $J$ retain underlying sets. Every function $DX\to Y$ is continuous, and every function $X\to JY$ is continuous, giving the adjunction. Its induced monad and comonad are idempotent, since repeated discrete topology or indiscrete topology construction changes nothing.