Zermelo's categoricity theorem

Zermelo's categoricity theorem was proven by Ernst Zermelo in 1930. It states that all models of a certain second-order version of the Zermelo-Fraenkel axioms of set theory are isomorphic to a member of a certain class of sets.

Statement
Let $$\mathrm{ZFC}^2$$ denote Zermelo-Fraenkel set theory, but with a second-order version of the axiom of replacement formulated as follows:


 * $$\forall F\forall x\exists y\forall z(z\in y \iff \exists w(w\in x\land z = f(w)))$$

, namely the second-order universal closure of the axiom schema of replacement. p. 289 Then every model of $$\mathrm{ZFC}^2$$ is isomorphic to a set $$V_\kappa$$ in the von Neumann hierarchy, for some inaccessible cardinal $$\kappa$$.

Original presentation
Zermelo originally considered a version of $$\mathrm{ZFC}^2$$ with urelements. Rather than using the modern satisfaction relation $$\vDash$$, he defines a "normal domain" to be a collection of sets along with the true $$\in$$ relation that satisfies $$\mathrm{ZFC}^2$$. p. 9

Related results
Dedekind proved that the second-order Peano axioms hold in a model if and only if the model is isomorphic to the true natural numbers. pp. 5–6 p. 1 Uzquiano proved that when removing replacement form $$\mathsf{ZFC}^2$$ and considering a second-order version of Zermelo set theory with a second-order version of separation, there exist models not isomorphic to any $$V_\delta$$ for a limit ordinal $$\delta>\omega$$. p. 396