What constitutes a partial play doesn't depend on $\alpha$. And $\kappa$ is assumed to have cofinality strictly larger than $\aleph_1$. So he means: define $\sigma$ applied to a partial play to just be some ordinal below $\kappa$ that is above all the outputs of the $\aleph_1$-many strategies $\sigma_\alpha$.

What constitutes a partial play doesn't depend on $\alpha$. And $\kappa$ is assumed to have cofinality strictly larger than $\aleph_1$. So he means: define $\sigma$ applied to a partial play to just be some ordinal below $\kappa$ that is above all the outputs of the $\aleph_1$-many strategies $\sigma_\alpha$.

(And it is easy to see that for any fixed $\alpha$, if $\sigma_\alpha$ is a winning strategy for II in the game $G_\alpha$, then so is any function from partial runs into $\kappa$ that is $\ge \sigma_\alpha$.)