The following amazing connection is a special case of a theorem by Sato Kentaro and another theorem by Normal Perlummter.

**Theorem:** The following are equivalent for regular $\kappa$:

*(i)* For every function $f: \kappa\rightarrow\kappa$, there is some $\alpha\lt\kappa$ such that $f``\alpha\subseteq\alpha$ and there is some $j: V\prec M$ with critical point $\alpha$, and $V_{j^2(f)(j(\alpha))}\subseteq M$.

*(ii)* For every function $f: \kappa\rightarrow\kappa$, there is some $\alpha\lt\kappa$ such that $f``\alpha\subseteq\alpha$ and there is some $j: V\prec M$ with critical point $\alpha$, and $M^{j(f)(\alpha)}\subseteq M$.

*(iii)* For every $rank(S)=\kappa$, there is some $\mathfrak M$,$\mathfrak N\in S$, and a $j: \mathfrak M\prec\mathfrak N$.

*Proof.* $(i)\leftrightarrow (iii)$ follows from a theorem in Sato Kentaro's *Double helix in large large cardinals and iteration of elementary embeddings*, and $(ii)\leftrightarrow (iii)$ from Norman Perlmutter's *The large cardinals between supercompact and almost-huge*.◼

Another amazing theorem is this:

**Theorem:** The following are equivalent:

*(i)* For every $\gamma$, there is some $j: V\prec M$ with critical point $\kappa$, and $V_{j(\gamma)}\subseteq M$.

*(ii)* For every $\lambda$, there is some $j: V\prec M$ with critical point $\kappa$, $M^{\lambda}\subseteq M$ and $V_{j(\kappa)}\subseteq M$.

*(iii)* $\kappa$ is extendible.

*Proof.* $(i)\leftrightarrow (iii)$ follows from a theorem in Sato Kentaro's *Double helix in large large cardinals and iteration of elementary embeddings*, and $(ii)\leftrightarrow (iii)$ from Konstantinos Tsaprounis' *Elementary chains and $C^{(n)}$-cardinals*.◼

These all highlight the shocking connection between strongness and extendibility.