If $\kappa$ is a $\lambda$-supercompact cardinal, then there exists a normal fine measure on $P_\kappa(\lambda)$. In Jech, Set Theory, we read that a function representing the ordinal $\lambda$ of the ultrapower is $f:P\mapsto ot(P)$ (order type).
He suggests to use the following: if $j$ is the elementary embedding of $V$ in the ultrapower and $G=\{j(\gamma):\gamma<\lambda\}$, then $[f]=jf(G)$ if $f$ is a function $f:P_\kappa(\lambda)\longrightarrow V$.
I am stuck in finding how to use these two facts. Can someone help me? Thank you in advance.
Let $U$ be a normal, fine measure on $\mathcal{P}_{\kappa}(\lambda)$, let $M := \mathrm{Ult}(V; U)$ be the ultrapower of $V$ by $U$ (which we identify with its transitive collapse) and let $$ j \colon V \to M, x \mapsto [c_x] $$ be the canonical elementary embedding.
Step 1. Let $d \colon \mathcal P_{\kappa}(\lambda) \to \mathcal P_{\kappa})(\lambda), X \mapsto X$ be the identity function. By Lemma 20.13 we have that $$[d] = j"\lambda := \{ j(\gamma) \mid \gamma \in \lambda \} = G.$$
Step 2. Note that $$ \begin{align*} (jf)(j"\lambda) &= [c_f](j"\lambda) \\ &= [c_f]([d]). \end{align*} $$ Now, by Łoś' Theorem, $$ \begin{align*} [f] = (jf)(j" \lambda) & \iff [f] = [c_f]([d]) \\ & \iff \{ X \in \mathcal P_{\kappa}(\lambda) \mid f(X) = c_f(X)(d(X)) \} \in U \end{align*} $$ But $\{ X \in \mathcal P_{\kappa}(\lambda) \mid f(X) = c_f(X)(d(X)) \} = \mathcal P_{\kappa}(\lambda)$ is certainly in $U$. Q.E.D.
Can you continue from here?