Simple examples of ambidextrous adjunctions and their frobenius monads I am looking for some simple examples of ambidextrous adjunctions.  There are two (I think) given here.  I am not sure what I want to specify as "simple", except to say that my first categories of interest were FinCat and Set (for which there is no ambidextrous adjunction).  I'm sure anything would help.
 A: These arise in the setting of Lawvere's Axiomatic Cohesion as "quality types".
To start with an example, consider the category of "bouquets", which consist of a set $F$ of "flowers" and a set $P$ of "pots" and a function $i : F \to P$ assigning each flower to its pot, and a section of $i$ which selects a particularly pretty flower in each pot. There is a natural functor $p : \textbf{Bouquet} \to \textbf{Set}$ sending a bouquet to its set of pots, and this has a left adjoint $e : \textbf{Set} \to \textbf{Bouquet}$ sending a set $X$ to the bouquet with one particularly pretty flower in each of its $X$ pots. But since morphisms of bouquets preserve the particularly pretty flowers, $e$ is also a right adjoint of $p$ (which is worth checking). So this is an example of an ambidextrous adjunction.
This is an example of what Lawvere calls the "canonical intesive quality of a cohesive topos", but which we could call the "category of infinitesimal spaces". A cohesive topos (over sets, for simplicity) is meant to be a topos of spaces whose points "cohere", or are "glued together". The axioms of such a topos concern the global sections functor $\Gamma$ which takes the points of a space, its left adjoint $\Delta$ which includes sets as discrete spaces. We then ask for a right adjoint to $\Gamma$ which includes sets as codiscrete spaces, and a further left adjoint to $\Delta$ called $\pi_0$ which takes the set of connected components of a space. There's some extra stuff but this is all we need.
We can see that $\pi_0$ takes connected components if we remember that maps in $\mathcal{E}$ are supposed to preserve "cohesion", and for that reason cannot send two points in a cohesive clump to two different points in a discrete space.
As an example, let $\mathcal{E}$ be the topos of reflexive graphs. Then $\Gamma$ takes the points of a graph, $\Delta$ includes the discrete graphs (with no edges but the reflexive ones), the codiscrete inclusion gives complete graphs, and $\pi_0$ takes the usual connected components.
We can define an infiniteismal space to be a space $X \in \mathcal{E}$ which has exactly one point in every connected component: $\pi_0 X \cong \Gamma X$. The intuition behind calling this "infinitesimal" is that this condition does not mean that $X$ is discrete, but it still lacks all "finite connection" because no two distinct points are in the same connected component.
Almost by definition, $\pi_0$ and $\Delta$ restrict to an ambidextrous adjunction on the full category of infinitesimal spaces (since $\pi_0 \cong \Gamma$ on these spaces). In the case of reflexive graphs, the infinitesimal spaces are bouquets, so this is the example I gave above. But there are plenty of examples of cohesive toposes, from the combinatorial to the analytic, each of which will give rise to its own ambidextrous adjunction in this way.
