Reflection by a beta-strong embedding

If a $\beta$-strong embedding has critical point $\kappa$ and the $\beta$-stable property $\Phi (\kappa )$ holds, then $\{\mu <\kappa :\Phi (\mu )\}$ is unbounded in $\kappa$. For each $\gamma <\kappa$, the target sees $\kappa$ as a witness between $\gamma$ and $j(\kappa )$; elementarity reflects a witness between $\gamma$ and $\kappa$.