My question seems far too basic to be unknown, but I could not find anything relevant...
Let $X$, $Y$ and $Z$ be compact connected metric spaces, and let $F \subset X \times Y$ and $G \subset Y \times Z$ be closed binary relations such that both projections are surjective. Suppose that they are also connected as topological spaces.
Is it true that $G \circ F$ is connected?
Probably I should also mention that a combinatorial version of this statement for connected graphs is known to be true.