In this post, Joel David Hamkins (Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2, URL (version: 2015-05-10): https://mathoverflow.net/q/206178) answers a questions about categorical theories extending second-order ZFC.

The way he obtains the categoricity of a theory T is by adding to ZFC2 axioms that make reference to models of T, i.e., ZFC2 +"there is a unique model of T, and no inaccessible cardinals above the size of that model", thus relying on Zermelo's theorem that the cardinality of any model of ZFC2 is an inaccessible cardinal.

My question is whether there is a way of making a theory categorical that doesn't rely on Zermelo's theorem, and thus might work for theories that were *not* extensions of ZFC2.

More specifically, I want to know whether the union of $\omega$-many $V_{\kappa}$ for $\kappa$ inaccessible (i.e. the countably-infinite union of domains of models of ZFC2) could be categorically characterized. Clearly, the union is not itself a model of ZFC2 since it has countable cofinality. Hence, we cannot use the "extension of ZFC2" method mentioned above.