Let there be three relations:
$g\subseteq D\times A\\ h\subseteq D\times B\\ R\subseteq(A\times B)\times C$
For the purposes of this post, the "composite of $R$ with $g$ and $h$," a subset of $D\times C$, is defined as
$R_{g,h}=\{(d,c)\in D\times C\mid \exists (d,x)\in g, (d,y)\in h \text{ such that } ((x,y),c)\in R\}$
I know the category Rel of binary relations uses sets as objects and relation as arrows, and relation composition (in the sense described here) as composition. Otherwise I have not worked in it, and to be honest it's a little bit weird.
I keep feeling like what's being described above is some simple categorical construct interpreted in Rel (like a pushout or some product), but it's hovering just outside of my grasp.
I'm faced with the given three arrows above, and the final arrow constructed with them, and I try to organize them in a diagram.
There should be an arrow from $D\to A\times B$ making a commutative diagram connecting $D,A,B$ and $A\times B$. Then $R$ gives this arrow to $C$, then naively I would look at the composition of arrows from $D$ to $A\times B$ to $C$, but I'm not sure this diagram is useful at all.
My two specific questions are:
What's the interpretation of this composition categorically (if there is one)?
Does it have a less bland name in the literature somewhere? (I would like to search for it.)
If I'm not mistaken, the relation $R_{g,h}$ is just $R\circ (g,h)$, where $(g,h) = g\times h \circ (\operatorname{id}_D, \operatorname{id}_D)$ is a relation from $D$ to $A\times B$.
The function / morphism $(\operatorname{id}_D, \operatorname{id}_D)$ is identified with its graph for the purpose of composing relations and $\times$ is just the Cartesian product applied to relations. So this makes sense categorically as well.
I don't know whether this has any special name.