all.
I have started reading "Distributors at Work" which is an introduction to distributors based on a course by Jean Bénabou: https://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf
On p. 2, they start by making an analogy between distributors and relations. Given sets $A$ and $B$, they say there is a correspondance between relations from $A$ to $B$ and functions $f:A$ $\rightarrow$ $\mathcal{ P}(B)$, which is clear. They then say that given Posets $A$ and $B$, one can also define relations from $A$ to $B$ as monotone maps from $A$ to $\downarrow B$ where this latter means "the poset of downward closed subsets of $B$ ordered by inclusion".
Now I am not able to prove this last claim. The last thing I tried was to use some kind of currying: if we regard relations as functions from $A \times B \rightarrow \{ False,True\}$, we have a correspondance with functions $F: A \rightarrow (B \rightarrow \{False,True\})$. If we consider that the functions $f:B \rightarrow \{False,True\}$ are monotone, then they correspond to downward closed sets. But I don't see how to make this work so we would have a correspondance with any possible relation.
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It think you misread the relevant section. Bénabou does not claim that monotone maps $A \rightarrow \downarrow B$ are in one-to-one correspondence with all possible set-theoretic relations between $A$ and $B$ (which wouldn't make much sense).
Instead, he defines a new notion, altogether: the notion of "relation between two posets $A$ and $B$". These are, by definition the monotone maps $A \rightarrow \:\downarrow\! B$. He then remarks that this newly defined notion of relation between posets also coincides with the notion of monotone map $B^{op} \times A \rightarrow 2$ where $2$ denotes the two-element lattice. This, you should be able to prove using a currying-style argument, i.e. using the Cartesian Closed structure of the category of posets and monotone maps.
Bénabou makes the pedagogical point that this looks analogous to how, if we wanted to, we could define set-theoretic relations between two sets $A,B$ as maps $A \rightarrow \mathcal{P}(B)$. And indeed, if we regard two sets $A,B$ as discrete posets, then $\mathcal{P}(B)$ is just $\downarrow\!B$, so the newly defined notion of relation and the set-theoretic notion of relation between $A$ and $B$ coincide.