(I have edited it. The previous version was with errors.)
Let $A$ be a set.
Let $\pi_0$, $\pi_1$ be projections from $A\times A$.
Let $F_0$, $F_1$, $G_0$, $G_1$ be binary relations on $A$.
Let $\Phi_A$ be the maximal binary relation in $(A\times A)\times(A\times A)$ such that $\pi_0\circ\Phi_A\subseteq F_0\circ\pi_0$ and $\pi_1\circ\Phi_A\subseteq F_1\circ\pi_1$.
Let $\Phi_B$ be the maximal binary relation in $(A\times A)\times(A\times A)$ such that $\pi_0\circ\Phi_B\subseteq G_0\circ\pi_0$ and $\pi_1\circ\Phi_B\subseteq G_1\circ\pi_1$.
Prove (or disprove) that $\Sigma=\Phi_B\circ\Phi_A$ is the maximal binary relation on $A$ such that $\pi_0\circ\Sigma\subseteq G_0\circ F_0\circ\pi_0$ and $\pi_1\circ\Sigma\subseteq G_1\circ F_1\circ\pi_1$.