Let $\Phi\subseteq (A\times A)\times(A\times A)$, $F_0,F_1\subseteq A\times A$ (for some set $A$) be binary relations.
I will denote $\pi_0$ and $\pi_1$ the projections of a cartesian product of sets.
Do inequalities $\pi_0\circ\Phi \subseteq F_0\circ\pi_0$ and $\pi_1\circ\Phi \subseteq F_1\circ\pi_1$ imply existence of a binary relation $\Psi\supseteq\Phi$ (we also require $\Psi\subseteq (A\times A)\times(A\times A)$) such that $\pi_0\circ\Psi = F_0\circ\pi_0$ and $\pi_1\circ\Psi = F_1\circ\pi_1$?
If yes, is this $\Psi$ unique?
Remark: For binary relations $f$ and $g$ I define the binary relation $g\circ f$ by the formula:
$(x,z)\in g\circ f \Leftrightarrow \exists y: ((x,y)\in f\wedge(y,z)\in g)$.