Let $F$ and $G$ be two endofunctors of a category $\mathcal C$ such that $(F,G, \eta, \epsilon)$ is an adjunction, i.e. I denote the unit by $\eta$ and the counit by $\epsilon$.
Then we may form two different directed systems in $\mathcal C$:
1) $C\to GFC \to G^2F^2C\to \cdots$
Here the first arrow is $\eta_C$, then $G\eta_{FC}$, then $G^2\eta_{F^2C}$, etc. So, the long arrow $C\to G^nF^nC$ is the unit of the adjunction $(F^n,G^n)$ obtained by composing the adjunction $(F,G)$ $n$ times with itself.
2) $C\to GFC \to GFGFC\to \cdots$
Here the first arrow is $\eta_C$, then $\eta_{GFC}$, then $\eta_{GFGFC}$, etc. Observe that we don't actually need the condition of the functors being endofunctors for this system to make sense.
Can we somehow compare these two systems? If moreover the colimits of these systems exist, can we compare those? Is there a way to make sense of the sentence "we get an adjunction $(F^\infty, G^\infty)$"? Is there some insight to be gained from the fact that we're iterating the monad associated to the adjunction, so we're not using the whole information that the adjunction contains?
Of course one can consider something similar with the functor $FG$ and using the counit instead of the unit.
This is a generalization of this question.