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 . 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).