Language: Bi-sorted FOL, upper cases for classes, lower for elements.
Primitives: equality '='; rudimentary membership '$\in' $'; Ordered pairing '$(,)$'. The first between any variables, the second from lower cases to upper, the third is a total function from lower cases to lower cases.
Axioms: bi-sorted ID, and:
Extensionality: $\forall m \, (m \in' X \iff m \in' Y) \implies X=Y$
Comprehension: $\exists X \forall y \, (y \in' X \iff \phi)$; where "$X$" doesn't occur in formula $\phi $.
Define: $X=\{y \mid \phi\}' \iff \forall y \, (y \in' X \iff \phi)$
Ordered pairing: $(a,b)=(c,d) \implies (a=c \land b=d)$
Existence: $\exists \operatorname {individuals} x_1,..,x_n: x_1 \neq ... \neq x_n$
Where: $\operatorname {individual}(x) \iff \neg \exists a \exists b: x=(a,b)$
Call the resulting system: Rudimentary Theory of Relations "$\sf RTR$", and the language: Rumdimentary Language of Relations "$\sf RLR$".
Now this language can capture the notions of set and membership spoken about in Set Theory!
Membership "$\in$" is definable as: $$\begin {align*} A \in B \iff & \exists b: [b \in' B \lor \exists x ((x,b) \in'B)] \ \land \\ & A=\{a \mid (a,b) \in' B \}' \big{)} \end {align*}$$
Sets are elements of the cumulative hierarchy of classes.
Now $\in$ so defined is non-extensional, yet this doesn't affect the standard developement of Set Theory, since $\sf Z + Collection -Ext.$ can interpret $\sf ZFC$, and so any set theory extending $\sf ZFC$ can be formalized in rudimentary langauge of relations.
I think a similar approach can work for Category theory.
In my opion the concepts of rudimentary membership and ordered pairing seems to be more trivial then epislon membership of Set Theory, that's why formalizing set theory in that language is (in my opinion) a kind of conceptual reduction.
This entails (in my opinion) that the Foundation of Mathematics is neither Set Theory nor Category theory.
Mathematics to be founded in the rudimentary language of relations.
Had this line of development been worked out before?