# Normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$

I am looking at fine normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$ and I have seen that there are two (seemingly) different formulations of normality, in terms of regressive (sometimes also called choice") functions.

Note that I am aware of the usual definition of normality in terms of diagonal intersections.

In what follows, fix (infinite) cardinals $\kappa<\lambda$, and assume that $\mathcal{U}$ is a $\kappa$-complete fine ultrafilter on $\mathcal{P}_{\kappa}(\lambda)$.

One formulation of normality is the one in Jech's book (Ch. 20, Definition 20.12). Namely, $\mathcal{U}$ is normal if and only if for every function $f:\mathcal{P}_{\kappa}(\lambda)\longrightarrow\lambda$ such that $f$ is regressive (i.e., $f(x)\in x$) for $\mathcal{U}$-almost all $x$, there is some $\alpha<\lambda$ such that $f(x)=\alpha$ for $\mathcal{U}$-almost all $x$. Note that this definition talks about something happening for $\mathcal{U}$-almost all $x$", i.e., on a set belonging to the ultrafilter.

The other (apparently equivalent) formulation of normality is given in Kanamori's book (Ch. 22, Exercise 22.6). There, $\mathcal{U}$ is normal if and only if for every function $f:\mathcal{P}_{\kappa}(\lambda)\longrightarrow\lambda$ such that $f$ is regressive (i.e., $f(x)\in x$) on a $\mathcal{U}$-stationary set (where $X\subseteq\mathcal{P}_{\kappa}(\lambda)$ is $\mathcal{U}$-stationary if $X\cap Z\neq\varnothing$, for all $Z\in\mathcal{U}$), there is some $\alpha<\lambda$ such that $f(x)=\alpha$ on a $\mathcal{U}$-stationary set.

My question is about the difference in these two formulations, and we are they equivalent.

In particular, I guess that something has to be said about sets belonging to the ultrafilter $\mathcal{U}$ vs. sets being $\mathcal{U}$-stationary. It is clear that every set that belongs to a filter $\mathcal{F}$ is certainly $\mathcal{F}$-stationary. On the other hand, in general, $\mathcal{F}$-stationary sets do not necessarily belong to the filter $\mathcal{F}$. For example, let $\mathcal{F}$ be the club filter on a regular uncountable cardinal $\kappa$, where one has disjoint stationary subsets of $\kappa$.

Can someone please clarify this situation?