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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:

1. Two arbitrary relations $M_1,M_2$, that does not preserve anything.
2. Two arbitrary functions $M_1,M_2$, that does not preserve anything.
3. Two relations $M_1,M_2$ such that $(x,x')\in M_1\wedge(y,y')\in M_2\Rightarrow [(x,y)\in R\Rightarrow (x',y')\in R']$
4. Two functions $M_1,M_2$ such that $(x,y)\in R\Rightarrow (M_1(x),M_2(y))\in R'$
5. An arbitrary relation $\rho\subset R\times R'$
6. An arbitrary function $f\subset R\times R'$
7. $M_1$ 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).
8. $M_2\circ R\subseteq R'\circ M_1$
9. 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.

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:

1. Two arbitrary relations $M_1,M_2$, that does not preserve anything.
2. Two arbitrary functions $M_1,M_2$, that does not preserve anything.
3. Two relations $M_1,M_2$ such that $(x,x')\in M_1\wedge(y,y')\in M_2\Rightarrow [(x,y)\in R\Rightarrow (x',y')\in R']$
4. Two functions $M_1,M_2$ such that $(x,y)\in R\Rightarrow (M_1(x),M_2(y))\in R'$
5. An arbitrary relation $\rho\subset R\times R'$
6. An arbitrary function $f\subset R\times R'$
7. $M_1$ 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).
8. $M_2\circ R\subseteq R'\circ M_1$
9. 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.

By considering facts I've come to the conclusion that there are two obvious canonical morphisms in categories with binary relations $X\overset R\longrightarrow Y$ as objects. $\require{AMScd}$ \begin{CD} X @>M_1>> X'\\ @VRV V ?@VV R\,'V\\ Y @>>M_2> Y' \end{CD}

Both general relations and only functions are natural alternatives and a list with possible candidates are:

1. The weakest morphisms are relations consisting of two arbitrary relations $M_1,M_2$, that does not preserve anything.
2. The relations (functions) constructed from arbitrary relations $\rho\subset R\times R'$ preserves the simple bi-graph structure, since $(x,x')\in M_1\wedge(y,y')\in M_2\wedge (x,y)\in R\Rightarrow (x',y')\in R'$ if $(x,x')\in M_1\Leftrightarrow [x\in \text{Coim}\,\rho \Rightarrow x'\in \text{Im}\,\rho]$, etc.
3. $M_1$ 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 $M_1^{op}$. (3 implies 2).
4. Commutative diagram. $($If $M_1$ and $M_2$ are functions all alternatives except 1 gives commutative diagrams$)$.

The canonical morphisms as I see it are either using functions and commutative diagrams or relations defined as in 2.

Relations can be used in the sense of investigate a problem in an other category using an adequate functor, as it seems to me. This is the essence of my answer on Mathematical structures and also occurs here and there.

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:

1. Two arbitrary relations $M_1,M_2$, that does not preserve anything.
2. Two arbitrary functions $M_1,M_2$, that does not preserve anything.
3. Two relations $M_1,M_2$ such that $(x,x')\in M_1\wedge(y,y')\in M_2\Rightarrow [(x,y)\in R\Rightarrow (x',y')\in R']$
4. Two functions $M_1,M_2$ such that $(x,y)\in R\Rightarrow (M_1(x),M_2(y))\in R'$
5. An arbitrary relation $\rho\subset R\times R'$
6. An arbitrary function $f\subset R\times R'$
7. $M_1$ 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).
8. $M_2\circ R\subseteq R'\circ M_1$
9. 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.

By considering facts I've come to the conclusion that there are two obvious canonical morphisms in categories with binary relations $X\overset R\longrightarrow Y$ as objects. $\require{AMScd}$ \begin{CD} X @>M_1>> X'\\ @VRV V ?@VV R\,'V\\ Y @>>M_2> Y' \end{CD}

Both general relations and only functions are natural alternatives and a list with possible candidates are:

1. The weakest morphisms are relations consisting of two arbitrary relations $M_1,M_2$, that does not preserve anything.
2. The relations (functions) constructed from arbitrary relations $\rho\subset R\times R'$ preserves the simple bi-graph structure, since $(x,x')\in M_1\wedge(y,y')\in M_2\wedge (x,y)\in R\Rightarrow (x',y')\in R'$ if $(x,x')\in M_1\Leftrightarrow [x\in \text{Coim}\,\rho \Rightarrow x'\in \text{Im}\,\rho]$, etc.
3. $M_1$ and $M_2$ are such that $M_2\circ R\circ M_1^{op}=R'$, that is the diagram is commutative for revered arrow with $M_1^{op}$. (3 implies 2).
4. Commutative diagram. $($If $M_1$ and $M_2$ are functions all alternatives except 1 gives commutative diagrams$)$.

The canonical morphisms as I see it are either using functions and commutative diagrams or relations defined as in 2.

Relations can be used in the sense of investigate a problem in an other category using an adequate functor, as it seems to me. This is the essence of my answer on Mathematical structures and also occurs here and there.

By considering facts I've come to the conclusion that there are two obvious canonical morphisms in categories with binary relations $X\overset R\longrightarrow Y$ as objects. $\require{AMScd}$ \begin{CD} X @>M_1>> X'\\ @VRV V ?@VV R\,'V\\ Y @>>M_2> Y' \end{CD}

Both general relations and only functions are natural alternatives and a list with possible candidates are:

1. The weakest morphisms are relations consisting of two arbitrary relations $M_1,M_2$, that does not preserve anything.
2. The relations (functions) constructed from arbitrary relations $\rho\subset R\times R'$ preserves the simple bi-graph structure, since $(x,x')\in M_1\wedge(y,y')\in M_2\wedge (x,y)\in R\Rightarrow (x',y')\in R'$ if $(x,x')\in M_1\Leftrightarrow [x\in \text{Coim}\,\rho \Rightarrow x'\in \text{Im}\,\rho]$, etc.
3. $M_1$ 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 $M_1^{op}$. (3 implies 2).
4. Commutative diagram. $($If $M_1$ and $M_2$ are functions all alternatives except 1 gives commutative diagrams$)$.

The canonical morphisms as I see it are either using functions and commutative diagrams or relations defined as in 2.

Relations can be used in the sense of investigate a problem in an other category using an adequate functor, as it seems to me. This is the essence of my answer on Mathematical structures and also occurs here and there.