I've run into the claim that $\diamondsuit_\kappa$ is equivalent to the (weaker looking) principle $\diamondsuit_\kappa^*$: That there's a sequence of $X_\alpha$ ($\alpha<\kappa$) with $X_\alpha\subseteq\mathcal{P}(\alpha)$, $|X_\alpha|\leq\alpha$, and for any $X\subseteq\kappa$, the set $\{\alpha<\kappa\mid X\cap\alpha\in X_\alpha\}$ is stationary in kappa.
It's pretty obvious why $\diamondsuit_\kappa$ would imply $\diamondsuit_\kappa^*$. But I can't see why the other direction should hold. Anyone have a quick answer to this question?