$ZF$ 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: $\forall x \forall M: x\epsilon M \Leftrightarrow x \varepsilon M$? 



Is any axiom system for sets categorical?
Of course not, since you can always take a permutation of $M$ and redefine $\in$ as the transport of structure defined by the permutation.


But even if you mean "up to isomorphism", the answer is still negative. Since given a model of set theory, by usual compactness+Lowenheim-Skolem arguments we can produce a non-isomorphic model with the same cardinality (so we can assume, without loss of generality that they have the same underlying set).


If you allow second-order axioms, then models of the form $V\_\kappa$ where $\kappa$ is a strongly inaccessible cardinal are exactly the models which satisfy the second-order replacement axiom, and then we can take $L\_\kappa$ and it is the unique model (up to a unique isomorphism) which satisfies the second-order axioms of $\sf ZFC$ and $V=L$.


But of course, this requires large cardinals, and second-order logic has been called "set theory in wolf's clothing" before.


