This is the continuation of part 1 of this question, where all useful definitions and notations are given. J. D. Hamkins answered question 1 in the first part, proving that there can be no injection of the proper class $\mathrm{On}$ into the proper class $W$, so that we now know all about the six injection possibilities between $\mathrm{On}$, $W$ and $V$.

This part (part 2) is about the answer to my question Bijective-equivalent collections of proper classes in set theory given by Ali Enayat. He proved that there exists a model of $\mathrm{NBG}$ where the proper class $P(\mathrm{On})$ is such that:

**(i)** Evidently $\mathrm{On}$ injects into $P(\mathrm{On})$ that injects into $P(P(\mathrm{On}))=V$;

**(ii)** But $V=P(P(\mathrm{On}))$ does not inject into $P(\mathrm{On})$, that itself does not inject into $\mathrm{On}$, so that the Proper Cardinal level $P(\mathrm{On})^*$ is distinct from both $\mathrm{On}^*$ and $V^*$;

**(iii)** Moreover there can be no injection from $P(\mathrm{On})$ into $W$, because we could chain such an injection with an injection from $\mathrm{On}$ into $P(\mathrm{On})$ and build an injection from $\mathrm{On}$ into $W$, which is impossible. Hence $W^*$, which is already distinct from the distinct proper Cardinal levels $\mathrm{On}^*$ and $V^*$, is also distinct from $P(\mathrm{On})^*$.

Then we see that Ali Enayat's model of $\mathrm{NBG}$ provides a case with (at least) four distinct Proper Cardinal levels.

**Question 2:** *Is it possible to build an injection from $W$ into $P(\mathrm{On})$ ?*

Concerning surjections, we have that:

**(i)** $V$ surjects onto $P(\mathrm{On})$ that itself surjects onto $\mathrm{On}$;

**(ii)** $V$ surjects onto $W$ that itself surjects onto $\mathrm{On}$.

**Question 3:** Is it possible to build a surjection from $P(\mathrm{On})$ onto $W$, or a surjection from $W$ onto $P(\mathrm{On})$, so that $\mathrm{On}$, $P(\mathrm{On})$, $W$ and $V$ are linearly ordered by surjection ?

**Question 4:** Is it possible to have more than four distinct Proper Cardinal levels in $\mathrm{NBG}$?