Let $T$ be a set of sentences in some first-order language $L$, and assume $T$ is negation-complete, so that for any sentence of $\phi\in L$, either $T\vdash\phi$ or $T\vdash\lnot\phi$. Assume as well that $T$ has a categorical-in-cardinality-$\kappa$ axiomatization $\Gamma$, i.e. assume there exists a set of sentences $\Gamma$ expressed in $L$ such that (1) the deductive closure of $\Gamma$ is $T$, and (2) all models of $\Gamma$ with cardinality $\kappa$ are isomorphic.
Let $\Omega$ be an arbitrary axiomatization of $T$. Does it follow that $\Omega$ must also be categorical in $\kappa$?
I don't know much logic beyond the definitions, so I don't have a supply of examples of $\kappa$-categorical theories to test the conjecture on. And I don't see immediately why it would be impossible for one axiomatization to be $\kappa$-categorical without all of them being so.