In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem:

"Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial elementary embedding $j:$$V$$\rightarrow$$V[G]$ [$V$$\vDash$$ZFC$--my comment].

They note:

"Attribution for this... theorem is not clear to us. It may have been known to Woodin, and Matt Foreman mentioned to the first author that he had discussed a version of it with Mack Stanley and Sy Friedman in the 1980's, but their proof was different from ours here and their result unpublished$.^{2}$"

Here is their footnote [2]:

"Part of their focus was reportedly on the extent to which the result generalized to class forcing. For example, they considered the case of class forcing extensions by amenable class forcings. Foreman mentioned that Woodin has an example of forcing using a class version of non-stationary tower forcing where $j:$$V$$\rightarrow$$V[G]$, but $V[G]$ does not have $ZFC$ for for the predicate $V$...."

If $\lnot$($V$$\vDash$ZFC), what is the theory that $V$ is a model of? Also, in this result of Woodin's, what is the generic object $G$?

In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem:

"Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial elementary embedding $j:$$V$$\rightarrow$$V[G]$ [$V$$\vDash$$ZFC$--my comment].

They note:

"Attribution for this... theorem is not clear to us. It may have been known to Woodin, and Matt Foreman mentioned to the first author that he had discussed a version of it with Mack Stanley and Sy Friedman in the 1980's, but their proof was different from ours here and their result unpublished$.^{2}$"

Here is their footnote [2]:

"Part of their focus was reportedly on the extent to which the result generalized to class forcing. For example, they considered the case of class forcing extensions by amenable class forcings. Foreman mentioned that Woodin has an example of forcing using a class version of non-stationary tower forcing where $j:$$V$$\rightarrow$$V[G]$, but $V[G]$ does not have $ZFC$ for for the predicate $V$...."

[Edit] Apparently, one has that when replaces 'set forcing' with 'class forcing' in Theorem 7, one can have a nontrivial elementary embedding $j:$$V$$\rightarrow$$V[G]$. This seems to contradict Kunen's inconsistency, but the comments made seem to say no. Why is this?