Suppose $A\dashv B\colon \cal H\to K$ are adjoint functors; if the comonad $AB$ has a right adjoint $T\colon\cal K\to K$ then $T$ is a monad in a natural way.

I would like to tell when "$T$ can be factored through $B$", i.e. $T=CB$ for some $B\dashv C$, $C\colon \cal H\to K$.

This is sort of a converse to the general statement that when you have $A\dashv B\dashv C$ then the comonad $AB$ has as right adjoint the monad $CB$.

Since my interest in this question comes from cartesian closed categories (I can provide details, but I don't think it would be useful for the moment), I tried to skim the Elephant looking for a clue, but I didn't find the right keyword... can you help me?