Suppose you in a more comprehensive framework has the result that U is the least set such that all axioms of NBG-- (NBG, minus extensionality and regularity), hold; replacement is here, as expected, obtained relative to the induced least class containing U and (membership class) E, and as well closed under complement, intersection, domain, Cartesian product with U, circular permutation and transposition.

It seems obvious that regularity holds for U.

Is it obvious that extensionality holds for U?

Edit:

The question(s) presuppose that NBG is consistent.

Edit 2: I changed the original V to U, so as to avoid confusion with the cumulative hierarchy.