$\mu$-Clubs of $P_{\kappa }(\lambda )$ : Paradise in heaven

Let $\mu <\kappa <\lambda$ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, $u(\kappa ,\lambda )$ is regular, where $u(\kappa ,\lambda )$ denotes the least size of a cofinal subset in $(P_{\kappa }(\lambda ),\subseteq )$, and cf$(\lambda )≠\mu$, then (a) the $\mu$-club filters on $P_{\kappa }(\lambda )$ and $P_{\kappa }(u(\kappa ,\lambda ))$ are isomorphic, and (b) the ideal dual to the $\mu$-club filter on $P_{\kappa }(\lambda )$ (and hence the restriction of the nonstationary ideal on $P_{\kappa }(\lambda )$ to sets of uniform cofinality $\mu$) is not $I_{\kappa ,\lambda }$-${\mathfrak{b}}_{u(\kappa ,\lambda )}$-saturated.