This is a problem in Hatcher's Algebraic topology.
Show that a finite graph product of finite groups has a free subgroup of finite index, by constructing a finite-sheeted covering space of $K\Gamma$ from universal covers of the mapping cylinders of $K\Gamma$.
I have absolutely no idea how the universal covers of the mapping cylinders look like when we don't know that the group homomorphisms are injective. And I have no idea how to construct this finite sheeted covering. Please help.
EDIT: $K\Gamma$ is the space made by placing any $BG\in K(G,1)$ at the place of $G$ and gluing the mapping cylinder of the induced morphism $BG\to BH$ at the place of edges $G\to H$. For detail see Hatcher 1B.7