Which are the natural morphisms between binary relations?

Question

Which properties should morphisms $\alpha$ between binary relations have?

$ (1) \qquad R \overset \alpha \longrightarrow R\,', \;R\subseteq X\times Y, \;R\,'\subseteq X'\times Y' $

Can those properties be expressed as relations $M_1$, $M_2$ below?

$ (2) \qquad M_1\subseteq X\times X', \;M_2\subseteq Y\times Y' $

If so, is the diagram commutative then? $\require{AMScd}$ \begin{CD} X @>M_1>> X'\\ @VRV V @VV R\,'V\\ Y @>>M_2> Y' \end{CD} Except for the commutative condition there is an other natural condition considering $R$ and $R\,'$ as (bipartite) graphs:

$ (3) \qquad (x,x')\in M_1\wedge(y,y')\in M_2\Rightarrow [(x,y)\in R\Rightarrow (x',y')\in R\,'] $

There is a counter-example below showing that the condition on the diagram being commutative, not in general imply $(3)$, if $M_1$ and $M_2$ not are functions.

What about if $M_1$ and $M_2$ are functions?

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Counter-example? $X=X'=Y=Y'=\mathbb{N}$ and $R=R\,'=M_1=M_2\wedge [(x,y)\in R \Leftrightarrow x<y]$

Answer 1

I think the most natural definition might just be

A morphism of relations $\alpha\colon R\to R'$ is a pair $(M_1,M_2)$ of relations $M_1\colon X\to X'$ and $M_2\colon Y\to Y'$ such that $R'\circ M_1=M_2\circ R$.

This definition at least makes the class of relations and morphisms into a category, and your diagram commutes by definition. You could of course restrict the hom sets to just functions making the diagram commute. I'm not sure how interesting these two categories are though.