Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 24 / 4 / ii / Solution

The left adjoint is discretization $D$: give the underlying set of $X$ the discrete topology. Write $J$ for indiscretization. Every function $DX\to Y$ is continuous, and every function $X\to JY$ is continuous. The common set of underlying functions supplies the natural bijectionThe discrete-indiscrete adjunction has induced monad $JD$, which assigns the indiscrete topology to the original set. Applying it twice gives the same space, and its multiplication is the identity on that set. Thus the adjunction is idempotent. Its dual comonad $DJ$, discretization, is idempotent as well.