The category with binary relations as objects I have reconsidered my ideas and remember how I thought ones upon a time. I will make a last try and delete if it doesn't work:
Set is the category where sets are objects and functions are morphisms and
Rel is the category where sets are objects and binary relations are morphisms.
I can think about two categories where functions are the objects, with pair of functions $(\alpha,\beta)$
$\require{AMScd}$
\begin{CD}
X @>\alpha>> X'\\
@VfV V @VVf'V\\
Y @>>\beta> Y'
\end{CD}
as morphism, with or without a condition of commutative diagrams ($f'\alpha=\beta f$). For relations these alternative do not preserve even the structure of relations as graphs in a natural way.
$$
\begin{CD}
X @>\alpha>> X'\\
@VR V\qquad \displaystyle ? V @VVR'V\\
Y @>>\beta> Y'
\end{CD}
$$
Is there a suitable condition on the diagram that preserves the graph-structure and make the pairs of relations $(\alpha,\beta)$ to morphisms?

The composition of relations $\alpha\subset X\times X'$ and $\alpha'\subset X'\times X''$ is defined
$(x,x'')\in\alpha'\alpha \Leftrightarrow \exists x'\in X':(x,x')\in\alpha \wedge (x',x'')\in \alpha'$.

You where right about my first attempt, but I hope that this is interesting enought, since many structures in mathematics in fact is just relations, and preserving those relation should be interesting.
 A: (Hurkyl + Ittay Weiss are right)
If u define Rel to have sets as objects and binary relations as arrows and you show this makes it a category, then u have (as for any category):

Say $\mathcal{C}$ is an arbitrary category (not necessarily small)
Define $\mathcal{\hat C}$ to be the category having as objects all $\mathcal{C}$-arrows and as arrows between two $\mathcal{\hat C}$-objects $f:A\longrightarrow A'$, $g:B\longrightarrow B'$ the pairs of $\mathcal C$-arrows $\phi_{AB}:A\longrightarrow B$, $\phi_{A'B'}:A'\longrightarrow B'$ such that the following diagram commutes
\begin{align}
A & \space\space\space\space\space\space\space\longrightarrow & B\\
\downarrow & & \downarrow \\
A' & \space\space\space\space\space\space\space\longrightarrow &B'
\end{align}
The composition of arrows is well defined since for $\mathcal{\hat C}$-objects $f:A\longrightarrow A'$, $g:B\longrightarrow B'$ and $h:C\longrightarrow C'$ and $\mathcal{\hat C}$-arrows (i.e. commuting $\mathcal{C}$-diagrams)
\begin{align}
A & \space\space\space\space\space\space\space\longrightarrow & B\\
\downarrow & & \downarrow \\
A' & \space\space\space\space\space\space\space\longrightarrow &B'
\end{align}
and
\begin{align}
B & \space\space\space\space\space\space\space\longrightarrow & C \\
\downarrow & & \downarrow \\
B' & \space\space\space\space\space\space\space\longrightarrow &C'
\end{align}
the combined $\mathcal{C}$-diagram commutes
\begin{align}
A & \space\space\space\space\space\space\space\longrightarrow & B & \space\space\space\space\space\space\space\longrightarrow & C \\
\downarrow & & \downarrow & & \downarrow \\
A' & \space\space\space\space\space\space\space\longrightarrow &B' & \space\space\space\space\space\space\space\longrightarrow &C'
\end{align}
thus buidling a composed $\mathcal{\hat C}$-arrow $\phi_{AC}:A\longrightarrow C$, $\phi_{A'C'}:A'\longrightarrow C'$
