Consider the three categories $\mathbf{Top}$ (of topological spaces with continuous maps), $\mathbf{Graph}$ (of simple graphs with maps that preserve or contract edges) and $\mathbf{Pros}$ (of pre-ordered sets with monotone maps). These three categories, albeit rather different in feel, share a common property: They come with two functors each from and to $\mathbf{Set}$, which we can call $\pi_0, D, U$ and $K$, that interact very nicely.
For each of these categories, $\pi_0$ maps an object to its set of connected components (defined in the sensible way for each category). The functor $D$ maps a set $M$ to the discrete object on that set in the category (i.e. for spaces, this will be the discrete topology on $M$, for graphs we have $M$ with no edges and for pre-orders, we have no nontrivial relations). As usual, $U$ is the forgetful functor to $\mathbf{Set}$ and, finally, $K$ makes a set $M$ into the "indiscrete" (or complete) object on that set (again defined in the sensible way in each case).
What makes these functors interesting is that all fit (well, almost fit) into an adjoint chain $$(\pi_0\dashv) D\dashv U \dashv K.$$
(The first adjunction is true for graphs and prosets, but false for topological spaces - it becomes true, however, if we restrict to the full subcategory of locally connected spaces instead).
The similarities do not end there - for each instance, all the composite functors $\pi_0\circ D, U\circ D, U\circ K$ give the identity on $\mathbf{Set}$, while $\pi_0\circ K$ collapses every nonempty set to a singleton. Moreover, in each case, the adjoint chain cannot be extended on either side. It is also not too difficult to extend this list of categories that have similar functors - for instance, the same thing works for the category of groupoids and the category of small categories (I think).
Is there an underlying principle at work here that makes all these categories behave so similarly? Or is it just a "fluke" somehow? Are there perhaps some examples where this adjoint chain can be extended beyond these four functors? I am grateful for any insights into this.