Pseudo-large cardinals

Let \(I\) be the least weakly inaccessible.

Let \(j\) be a total (not necessarily injective) map \(V \to V\) satisfying the following:

1. For \(\kappa, \lambda\), if \(\kappa \in \lambda\), then \(j(\kappa) \in j(\lambda)\) or \(j(\kappa) = j(\lambda)\).
2. If \(\kappa\) is regular but not weakly inaccessible, but \(I \in \kappa\), then \(j(\kappa) \in \kappa\) will be an irregular limit of regular cardinals.
3. If \(\kappa\) is weakly inaccessible, then \(j(\kappa) \in \kappa\) will be regular but not weakly inaccessible.

Then, \(\kappa\) is pseudo-inaccessible if there is some map \(j\) satisfying the above properties, and \(j(\kappa)\) is weakly inaccessible.