Taken out of ch II, Kunen. Need to show the following two versions of $\Diamond$ are equivalent:
$\Diamond_\kappa$: There are $A_\alpha \subset \alpha$ for $\alpha < \kappa$ such that for each $A \subset \kappa$ the set $\{\alpha < \kappa : A \cap \alpha = A_\alpha\} $ is stationary.
$\Diamond_\kappa'$: There are $A_\alpha \subset \mathcal P(\alpha)$ for $\alpha < \kappa$ such that each $|A_\alpha| \leq \alpha$ and for each $A \subset \kappa$, $\{\alpha < \kappa : A \cap \alpha \in A_\alpha\}$ is stationary.
$\Diamond_\kappa \Leftarrow \Diamond_\kappa'$ was shown on one of the lectures (for $\omega_1$ instead of $\kappa$) and it involved using $f: \omega_1 \leftrightarrow \omega \times \omega_1$ as a set-building function for the transition. Is the other direction more straightforward? Seems like it's possible to build sets of $\diamond_\kappa'$ from sets of $\diamond_\kappa$ by applying an easy transformation to them, but the condition $|A_\alpha| \leq \alpha$ seems tricky here.
Any help would be appreciated.