Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding the height of the cumulative hierarchy unspecified. 


So my question is if second-order ZFC can at least be said to be categorical with regard to its proper class models (i.e. having a unique proper class model up to isomorphism)? And in that sense is it also correct to say that ZFC2 has a unique 'largest' model (up to isomorphism), since the proper class models are 'larger' than the set models. 


I am assuming standard semantics for second order logic (e.g. not Henkin semantics).


[1] <http://web.mit.edu/arayo/www/puzzle.pdf>



Is second-order ZFC categorical with regard to its proper class models
The only second-order part that is needed for this is the second-order power set axiom. 


That is, if $M$ is any proper class model of (a weak fragment of) first-order ZFC and satisfies the second-order power set axiom, then $M$ contains all sets. The reason is that if we iterate the power set operation inside $M$, then we will construct the actual $V\_\alpha$'s, since $M$ is correct about power sets. And so every $V\_\alpha$ is contained in $M$ and so $M=V$. 


