$ZF$ define membership by conditions demanding the existence of some constructable right-side-terms $M $ ($x \in M$). Is it meaningsful to ask for a categorical axiom system here? Shouldn't it be prooved, that any two relations $\epsilon$ and $\varepsilon$ which satisfying the axioms allways fullfill: $\forall x \forall M: x\epsilon M \Leftrightarrow x \varepsilon M$?