A category of relations - or two different?

Question

Objects in the category Rel2 (my notation) are the relations $r\subseteq A\times B$, $r'\subseteq A'\times B'$ (the morphisms in Rel) and morphisms are pair of relations $\alpha\subseteq A\times A'$ and $\beta\subseteq B\times B'$ such that

$(1)\quad(a,a')\in\alpha\wedge(b,b')\in\beta\implies\big((a,b)\in r\implies(a',b')\in r'\big)$

or equivalently

$(1')\!\!\quad(a,a')\in\alpha\wedge(b,b')\in\beta\wedge(a,b)\in r\implies(a',b')\in r'$.

My question is if all relations $R\subseteq r\times r'$ define a morphism in Rel2, that is, if given $R$ there exist relations $\alpha\subseteq A\times A'$ and $\beta\subseteq B\times B'$ such that:

$((a,b),(a',b'))\in R\iff \big((a,a')\in\alpha\wedge(b,b')\in\beta\wedge(a,b)\in r\implies(a',b')\in r'\big)$?

---

Some context:

Suppose $A=B\times B$ and that $r\subseteq A\times B$ is the composition in a magma. Then the functions among the morphisms between two such objects defines magma morphisms $B\to B'$.

Suppose $B=\mathcal P(A)$ and that $r\subseteq A\times B$ is the relation $(a,S)\in r\iff a\in\overline{S}$ for some topology on $A$. Then the functions among the morphisms between two such objects define continuous functions $A\to A'$.

---

Edit:
It is certainly two different categories. Suppose all four sets $A,B,A',B'$ have two elements. Then the number of relations between two such sets is $2^{2\times 2}=16$ and therefore the number of different pairs $\alpha,\beta$ is $16\times 16=256$.

While the number of relations between two relations is $2^{16\times 16}=$ $115792089237316195423570985008687907853269984665640564039457584007913129639936$

Answer 1

$\require{AMScd}$I don't quite understand the question, but basically, I think you're trying to define a particular double category.

Objects. Sets

Arrows. Relations

Proarrows. Relations

Squares. We assume that each square has at most one filler, and that it has a filler iff the condition $$\quad(a,a')\in\alpha\wedge(b,b')\in\beta\implies\big((a,b)\in r\implies(a',b')\in r'\big)$$ is satisfied.

(Think of $\alpha$ and $\beta$ as arrows, and $r$ and $r'$ as pro-arrows.)

This looks like a potentially important idea, and I'd encourage you to keep thinking more about it. You may or may not find this post of mine relevant, or at least interesting.