The paper "Double helix in large large cardinals and iteration of elementary embeddings" by Kentaro Sato 2007 gives the following characterization of almost huge cardinals as lemma 5.4 (note that $\kappa$-normal ultrafilters of $\mathcal{N}_{\kappa_{n−1},\kappa_n}$ are usually called coherent sequences of ultrafilters on $\mathcal{P}_{\kappa_{n−1}} \mu$ for $\mu \lt \kappa_n$):
$\kappa$ is $n$-fold almost huge with targets $\kappa_1$,…,$\kappa_n$ iff $\kappa_1$,…,$\kappa_n$ are inaccessible and there is a $\kappa$-normal ultrafilter $A$ of $\mathcal{N}_{\kappa_{n−1},\kappa_n}$ such that
- $\{s \in \mathsf{D}(\mathcal{N}_{\kappa_{n−1},\kappa_n})|ot(s \cap \kappa_{i+1}) = \kappa_i \} \in A$ for $i \lt n−1$ where $\kappa_0=\kappa$ and,
- for any $f \in \mathrm{Ult}(\mathcal{N}_{\kappa_{n−1},\kappa_n})$ with $Im(f) \subset \kappa_{n−1}$, there is $\nu \lt \kappa_n$ such that $\{s \in \mathsf{D}(\mathcal{N}_{\kappa_{n−1},\kappa_n})|f(s) \lt ot(s \cap \nu)\} \in A$.
$\mathrm{Ult}(\mathcal{N}_{\kappa_{n−1},\kappa_n})$ is the class of functions with domain $\mathcal{P}_{\kappa_{n−1}} \kappa_n$ such that there is an ordinal $\mu_f \lt \kappa_n$ such that $f(s)=f(t)$ whenever $s \cap \mu_f = t \cap \mu_f$ (they can also be seen a functions $\overline{f}$ with domain $\mathcal{P}_{\kappa_{n−1}} \mu_f$). I claimed in this Mathoverflow answer that this, minus inaccessibility, is satisfied by a club of cardinals $\lambda \lt \kappa_n$ in place of $\kappa_n$. Why doesn't this hold for every $\lambda$ such that $\kappa_{n−1} \lt \lambda \lt \kappa_n$? A function $f \in \mathrm{Ult}(\mathcal{N}_{\kappa_{n−1},\kappa_n})$ is in $\mathrm{Ult}(\mathcal{N}_{\kappa_{n−1},\lambda})$ iff $\mu_f \lt \lambda$ and whether $f(s) \lt ot(s \cap \nu)$ depends only on $s \cap \nu$.
I first posted this on Mathoverflow, where it was ignored, but on further consideration I think that my confusion is not research-level so I have deleted it there and posted it here instead.
Best Answer
Let $j$ denote the ultrapower embedding by $A$ and suppose $\kappa \leq \lambda \leq j(g)(\kappa)$ for some $g : \kappa \to \kappa$ (for example, if $g(\alpha)$ is the successor cardinal of $\vert \alpha \vert$, then $j(g)(\kappa)$ is the successor cardinal of $\vert \kappa \vert$). Define $f : \mathcal{P}_\kappa \lambda \to \kappa$ by $f(s) = g( sup (s) \cap \kappa)$. Then $Im(f) \subseteq \kappa$ and $\mu_f = \kappa$ but $\{s \in \mathcal{P}_\kappa \lambda | f(s) \geq ot(s) \} \in A$, for the ordinals less than $\lambda$ are represented by functions $f_\zeta : \mathcal{P}_\kappa \lambda \to \kappa$ such that, for all $\zeta \lt \eta \leq \lambda$, $f_\zeta(s) \lt f_\eta(s)$ for $A$-almost all $s$, $f_\kappa (s) = sup(s \cap \kappa)$ (by lemma 3.6 of the linked paper) and $f_\lambda$ is $f$ as defined above. Thus $f$ is a counterexample to the last clause of the quoted characterization of almost-hugeness ultrafilters.
More generally, if $\lambda \leq j(g)(\xi)$ for $\kappa \leq \xi \lt \lambda$, define $f : \mathcal{P}_\kappa \lambda \to \kappa$ by $f(s) = g(f_\xi(s))$.