 the following commutative diagram: $$\begin{aligned}  \xymatrix{\mathcal{T}\ar[r]^{i_{m}}\ar[d]^{\pi_m}&\mathcal{T}^H_{_m}\ar[d]^{\pi_{_m}^H}\ar[r]^{{\Phi}^{H}_{_m}}&D\ar[d]^{\pi_D}\\ \mathcal{Z}_m\ar[r]^{i}&\mathcal{Z}^H_{_m}\ar[r]^{{\Phi}_{_{\mathcal{Z}_m}}^H}&D/\Gamma. }\end{aligned}$$ 