} \xymatrix{\mathcal{T}\ar[r]^{i_{m}}\ar[d]^{\pi_m}&\mathcal{T}^H_{_m}\ar[d]^{\pi_{_m}^H}\ar[r]^{{\Phi}^{H}_{_m}}&D\ar[d]^{\pi_D}\\ \mathcal{Z}_m\ar[r]^{i}&\mathcal{Z}^H_{_m}\ar[r]^{{\Phi}_{_{\mathcal{Z}_m}}^H}&D/\Gamma, }\end{aligned}$$ where $i$ is the inclusion map, ${i}_{_m}$ is a lifting map of $i\circ\pi_m$, $\pi_D$ is the covering map and ${\Phi}^{H}_{_m}$ is a lifting map of ${\Phi}_{_{\mathcal{Z}_m}}^H\circ \pi_{_m}^H$. In particular, $\Phi^H_{_m}$ is a continuous map from $\mathcal{T}^H_{_m}$ to $D$. We notice that the lifting maps ${i}_{_{\mathcal{T}}}$ and ${\Phi}^H_{_m}$ are not unique, but Lemma \[choice\] implies that there exists a suitable choice of $i_{m}$ and $\Phi_{_m}^H$ such that $\Phi=\Phi^H_{_m}\circ i_{m}$. We will fix the choice of $i_{m}$ and $\Phi^{H}_m$ such that $\Phi=\Phi^H_{_m}\circ i_m$ in the rest of the paper. Let us denote $\mathcal{T}_m:=i_{m}(\mathcal{T})$ and the restriction map $\Phi_m=\Phi^H_{_m}|_{\mathcal{T}_m}$. Then we also have $\Phi=\Phi_m\circ i_m$. 