Let $\kappa < \lambda$ be regular cardinals and let $A \subset \lambda$ be such that each $\alpha \in A$ has cofinality $\gamma < \kappa$. Then the following should hold:

$A$ is stationary iff $ \lbrace$$X \in P_{\kappa} (\lambda) : sup(X) \in A $ $\rbrace$ is stationary in $P_{\kappa} (\lambda)$

The direction from the right to the left is not hard, but despite looking harmless, the $\Rightarrow$ direction puzzles me now for a while.

It should be enough to show that for any club $C$ in $P_{\kappa} (\lambda)$ the set $\underset{\sim}{C}:=$ $\lbrace$ $\beta \in \lambda : \exists X \in C \quad sup(X) = \beta$ $\rbrace$ has a $\gamma$-closed, unbounded subset of $\lambda$ which would imply (by the assumption that each element of $A$ has cofinality $\gamma$) that $\underset{\sim}{C} \cap A \ne \emptyset$, witnessing the stationarity of $ \lbrace$$X \in P_{\kappa} (\lambda) : sup(X) \in A $ $\rbrace$. However all my attempts to show this were cumbersome and dissatisfactory.