[Here](https://dl.acm.org/doi/pdf/10.1145/3661814.3662137) \[1\]  we see a paper that describes when you can compose a directed container over distribution monads.  Here is their definition of the coreader comonad.

[![enter image description here][1]][1]

A directed container will have a kind of graph structure.  For instance, the nonempty list comonad is just a chain with a head.

What I am wondering about is what is the graph structure of the coreader comonad seen as a directed container?  In particular, I want to know if the graph structure resembles a complete lattice, and hence a dcpo.  What are the restrictions on the set $S$?  For instance, in the binary tree directed container, S is literally binary trees.  What kinds of shapes, $s$, admit a coreader comonad?  What are some examples of coreader commands where we are interested in interesting shapes $S$?  Can we just outright say that $S$ is the set of all dcpos?  Is it possible the positions $P(S) = \{ \top \}$ are the least upper bounds for each dcpo in $S$?

Edit

The coreader comonad is related to the [quantum reader monad](https://ncatlab.org/nlab/show/quantum+reader+monad).  This reader is then based on the complete lattice of [protection operators on a hilbert space](https://scholar.google.ca/scholar?hl=en&as_sdt=0%2C5&q=complete+lattice+if+projection+operators&oq=complete+lattice+if+projection+operator#d=gs_qabs&t=1726923601367&u=%23p%3DKHh1DRzOjbMJ).  There seems to be something non-trivial to my idea.


\[1\] Karamlou, Amin, and Nihil Shah. "No Go Theorems: Directed Containers That Do Not Distribute Over Distribution Monads." Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science. 2024.


  [1]: https://i.sstatic.net/19HsFki3.png