Categories with binary relations as objects 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.
 A: If you write your diagram in the right way:
$$\begin{CD}
X @<M^{co}_1<< X'\\
@VRV V @VV R\,'V\\
Y @>>M_2> Y'
\end{CD}$$
(where $M^{co}_1 \colon X' \rightarrow X$ is obtained from $M_1 \colon X \rightarrow X'$ by swapping domain with codomain)
and restate the second condition in the right way:
$$(x',x)\in M^{co}_1 \wedge (x,y)\in R \wedge (y,y')\in M_2 \Rightarrow (x',y')\in R\,'$$
then it will become obvious, that your condition says that the diagram is weakly commutative --- i.e. it is commutative up to a 2-morphism in $\mathit{Rel}$ (recall that $\mathit{Rel}$ has a 2-categorical structure):
$$M_2 \circ R \circ M_1^{co} \leq R'$$
Therefore, you have described two different constructions over a (2-)category. Which is "more natural", depends on your applications.

I do not want to go into unnecessary detail, but here is an abstract argument why (1) implies (2) if $M_1$ is a function. Let us assume that: $$M_2 \circ R = R' \circ M_1$$ Because $M_1$ is a function, it has a right adjoint relation $M_1^\mathit{co}$. Now, we may postcompose our expression with $M_1^\mathit{co}$ to obtain $M_2 \circ R \circ M_1^\mathit{co} = R' \circ M_1 \circ M_1^\mathit{co}$. However, since $M_1^\mathit{co}$ is right adjoint to $M_1$, there is evaluation $M_1 \circ M_1^\mathit{co} \leq \mathit{id}$. Thus:
$$M_2 \circ R \circ M_1^\mathit{co} = R' \circ M_1 \circ M_1^\mathit{co} \leq R' \circ \mathit{id} = R'$$
In fact, the above proof works in a more general context of any single-valued relation $M_1$. Moreover, this is the optimal general condition --- if $M_1$ is not single-valued, then one may easily find $R, R', M_2$ such that (1) is true, but (2) does not hold.
A: By reconsidering facts I've come to the conclusion that there might be several morphisms to be used in categories with binary relations $ R\subseteq X\times Y$, eventually depending on the interpretation of the objects. 
$\require{AMScd}$
\begin{CD}
X @>M_1>> X'\\
@VRV V ?@VV R\,'V\\
Y @>>M_2> Y'
\end{CD}
Alternative morphisms:


*

*Two arbitrary relations $M_1,M_2$, that does not preserve anything.

*Two arbitrary functions $M_1,M_2$, that does not preserve anything.

*Two relations $M_1,M_2$ such that $(x,x')\in M_1\wedge(y,y')\in M_2\Rightarrow
[(x,y)\in R\Rightarrow (x',y')\in R']$

*Two functions $M_1,M_2$ such that $(x,y)\in R\Rightarrow (M_1(x),M_2(y))\in R'$

*An arbitrary relation $\rho\subset R\times R'$

*An arbitrary function $f\subset R\times R'$

*$M_1$ and $M_2$ are such that $M_2\circ R\circ M_1^{op}=R'$, that is the diagram is commutative for reversed arrow with $M_1^{op}$. (7 implies 3).

*$M_2\circ R\subseteq R'\circ M_1$

*Commutative diagram. (If $M_1$ and $M_2$ are functions all alternatives except 1 gives commutative diagrams).


There are even some more, but perhaps also some of the conditions are equivalent.
In Abstract and Concrete Categories (p. 22) the category of binary relations $R\subseteq X\times X$ on a set $X$ is called Rel and have morphisms in accordance with 4.
