I am trying to understand the definition of a relational $\beta$-module as described here: https://ncatlab.org/nlab/show/relational+beta-module.
The definition given in section $2$ under the title "Bridge to a Concrete Description" is as follows:
A relational $\beta$-module is a set $S$ and a binary relation $\xi: \beta S \rightarrow S$ between ultrafilters on $S$ and elements of $S$ satisfying these requirements ( given by $(1)$ in the linked article),
Question: What does the inequality sign mean here?
I can't seem to find descriptions of this concept in the linked article or elsewhere.
As mentioned in some of the comments, those diagrams take place in the category of sets and relations, $\operatorname{Rel}$. $\operatorname{Rel}$ isn't "just" a category: it's enriched over posets. That is, there isn't just a set of relations between sets $A$ and $B$, there's a poset of them (ordered by $\subseteq$).
The inequalities in the diagrams mean that they aren't intended to be commutative diagrams. Instead, they mean that the composition of relations on the left of the $\leq$ sign is contained as a subset of the composition of relations on the right (rather than equal to).