 vertex groups of $\Gamma \mathcal{G}$ to conjugates of vertex groups of $\Phi \mathcal{H}$. Then there exist an automorphism $\alpha \in \mathrm{ConjP}(\Gamma \mathcal{G})$ and an isometry $s : \Gamma \to \Phi$ such that $\varphi \alpha$ sends isomorphically $G_u$ to $H_{s(u)}$ for every $u \in V(\Gamma)$.  It 