Let $ U_1$, $ U_2$, $V_1,V_2$ be finite sets. Let $F\in \Delta(U_1\times U_2)$ and $G\in \Delta(V_1\times V_2)$ be some given probabilities distributions on $U_1\times U_2$ and $V_1\times V_2$.
Given $F$ and $G$, there may or may not exist two functions (injections) $\Gamma_1: U_1\to V_1$ and $ \Gamma_2: U_2\to V_2 $ such that
$$F((u_1,u_2): \Gamma(u_1)=v_1, \Gamma_2(u_2)=v_2)=G(v_1,v_2)$$
What would be the necessary and sufficient condition on $F$ and $G$ such that mappings $F_1$ and $F_2 $ exist? Is there any literature on this problem?
I think the result might related to Hall's marriage theorem but don't have further clue about it.
Thanks.