Theorem

For $x \in X,y\in Y$ let $$\begin{align*} E_{x}&=\{ y\in Y:(x,y)\in E \}\\ E^{y}&=\{ x \in X:(x,y)\in E \} \end{align*}$$ If $E\in\mathscr{P}(=\mathscr{M}\otimes \mathscr{N})$, then $\forall x \in X,\forall y \in Y$ we have $$E_{x}\in\mathscr{N}\text{ and }E^{y}\in\mathscr{M}$$