User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-24509095-20140507175906/@comment-5150073-20140508112547

I think the best case is define $$\psi_\alpha(\psi_\alpha(\psi_\alpha(\cdots ))) = \psi_\alpha(\alpha)$$, i.e. index of $$\psi$$ denotes an associated level of diagonalizer inside it. So instead of write $$\psi(\psi_1(\Omega_2))$$ we write $$\psi_{\Omega}(\psi_{\Omega_2}(\Omega_2))$$. Probably that removes all offsets and inaccuracies.