Imagine we define a relational structure as an ordered pair, $(D,R)$, such that the set/class, $D$, is the domain, and the set/class, $R$, consists of relations of various arities over $D$. Now, let's define a category, RelStr, with (all possible) $(D,R)$s as its objects, and homomorphisms between such $(D,R)$s as its arrows.
(a homomorphism, $h:(D_1,R_1)\rightarrow(D_2,R_2)$, maps $D_1$ to $D_2$, $R_1$ to $R_2$, and for any ordered $n$-tupe of the elements of $D_1$, like $(d_1,d_2,...,d_n)$, and any member of $R_1$, like $r$, if $(d_1,d_2,...,d_n)\in r$, then $(h(d_1),h(d_2),...,h(d_n))\in h(r)$.)
Now, I wanted to know if anyone's aware of a place where such categories have been studied (even briefly). Based on my very sketchy search in the literature, when people talk about categories of relational structures, they usually fix a signature, $\sigma$, and take the objects of the category to be $\sigma$-structures. That's why I'm wondering if the category of relational structures, in the sense I just defined, has been of any interest to people.