User blog comment:MachineGunSuper/The Hierarchy Replacement Function/@comment-11227630-20171224020855

That's similar to this. If it's correctly defined, it works up to $$f_{\psi(\Omega_\omega)}$$; but if it's incorrectly defined, it may be as weak as $$f_{\varepsilon_0}$$.

Catching function is an extension that helps you understand ordinals beyond $$f_{\psi(\Omega_\omega)}$$. But I haven't found a way to define it without refering other ordinal notations and fundamental sequences.