User blog comment:PsiCubed2/How to make Deedlit's Mahlo-level notation more intuitive/@comment-30869823-20180808201546/@comment-35470197-20180818122544

@Fejfo

The supremum has nothing to do with the fixed point in this case. Emlightened refers to the simple fact that if you have \(\alpha = I(M^{\alpha})\), then \(\alpha\) is regular because it lies in the image of \(I\) by the assumption.