Cardinality of classes

I am trying to define an embedding whose range includes classes. Is there a coherent way of assigning "cardinality" to proper classes?

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Without assuming Von Neumann's axiom, or the axiom of choice, I know of no sensible way of making sense of this notion, as now we could have some proper classes that are "thinner" than others, or even incomparable. Of course, we could study models where this happens (for example, work in ZF, assume there is a strong inaccessible $\kappa$, and consider $V_\kappa$ as the universe of sets, and $Def(V_\kappa)$ in Gödel's sense (or even $V_{\kappa+1}$) as the collection of classes).
Just to drive the point home, if choice fails, it may not be possible to force it back. For example, one cannot force (local) choice over Gitik's model where all (well-ordered) cardinals have cofinality $\omega$. –  Andres Caicedo Mar 8 '12 at 14:54