For the category of functions, pairs of functions making commutative diagrams are the canonical morphisms $\alpha:f\rightarrow g$. For binary relations there is an alternative, to consider the relations as (bipartite) graphs with canonical morphisms that preserves the graph structure. $\require{AMScd}$ \begin{CD} X @>M_1>> X'\\ @VRV V @VV R\,'V\\ Y @>>M_2> Y' \end{CD} $(1)\quad$ $R\,'\circ M_1=M_2\circ R$

$(2)\quad$ $(x,x')\in M_1 \wedge (y,y')\in M_2 \Rightarrow [(x,y)\in R\Rightarrow (x',y')\in R\,']$

In general, $(1)$ do not imply $(2)$. Define $(x,y)\in S \Leftrightarrow x\leq y$: \begin{CD} \mathbb{N} @>S>> \mathbb{N}\\ @VSV V\# @VV SV\\ \mathbb{N} @>>S> \mathbb{N} \end{CD} The diagram is trivially commutative, but of course $x\leq x'\wedge y\leq y'\wedge x\leq y \nRightarrow x'\leq y'$.

The condition $(2)$ is a generalization of $(1)$ and they are equivalent in case of functions.

Both $(1)$ and $(2)$ meets the conditions for morphisms, but which is the "most natural"?

Do $(1)$ imply $(2)$ for functions $M_1,M_2$?

The latter alternative gives a category of (simple) mathematical structures, with objects $F(X)\longrightarrow X$, for some functor $F$ on **Rel**, with some interesting properties that I have tried to indicate here and there.

I asked similar questions on Mathematics. And a related question also on Mathematics.