 Countable-family diamond principle

At each $\alpha \in S$, a countable family $E_{\alpha }\subseteq \mathcal{P}(\alpha )$ is supplied. Every $X\subseteq \lambda$ has $X\cap \alpha \in E_{\alpha }$ on a stationary subset of $S$. The countable-family diamond equivalence shows that this apparently weaker prediction is equivalent to a single diamond guess at each index.