Uniformization of a binary relation

For $A\subseteq X\times Y$, a uniformization is a function $f$ on the projection of $A$ to $X$ such that $(x,f(x))\in A$ for every $x$ in that projection. It chooses one witness from every nonempty vertical section of $A$.

Under ${\mathsf{AD}}_X$, every relation $A\subseteq X\times X$ has a uniformization. Let the first moves of Players I and II be $x$ and $y$, and let II win when $x$ is outside the projection of $A$ or $(x,y)\in A$. Player I cannot have a winning strategy, so the first response of a winning strategy for II uniformizes $A$.