\begin{tikzcd}
\overline{U}_{J',\alpha'} \arrow[r] \arrow[d] & \overline{U}_{J,\alpha} \arrow[r] \arrow[d] & \overline{U} \arrow[d] \\
U_{J',\alpha'} \arrow[r] & U_{J,\alpha} \arrow[r] & U
\end{tikzcd}

\begin{tikzcd}
D = U(\mathbb{C})^{\sim} \arrow[rr] \arrow[dr, "\pi"'] & & \widetilde{U}(\mathbb{C}) \arrow[dl, "\hat\pi"] \\
 & U(\mathbb{C}) &
\end{tikzcd}

\begin{tikzcd}
D \arrow[r, "u"] \arrow[d, "p'"'] & D \arrow[d, "p''"] \\
D/\pi' \arrow[r, "\sim"'] & D/\pi''
\end{tikzcd}

\begin{tikzcd}
\pi \arrow[d, "\varphi"'] & \Gamma & B_{\pi,\Gamma} \arrow[d] & B_{\pi} \arrow[l] \arrow[d] & \widetilde{B}_{\pi} \arrow[l] \\
\pi' & \Gamma & B_{\pi',\Gamma} \arrow[d, no head] & B_{\pi'} \arrow[l] \arrow[d] & \\
 & & B_{\Gamma} & \widetilde{B}_{\Gamma} \arrow[l] &
\end{tikzcd}

\begin{tikzcd}
B_{\pi,\Gamma} \arrow[r, no head] \arrow[d, no head] & B_{\pi,\Gamma_i} \arrow[d, no head] & B_{\pi} \arrow[l] \arrow[d, no head] \\
B_{\pi',\Gamma} \arrow[r, no head] \arrow[d, no head] & B_{\pi',\Gamma_i} \arrow[d, no head] & B_{\pi'} \arrow[l] \arrow[d, no head] \\
B_{\Gamma} \arrow[r, no head] & B_{\Gamma_i} \arrow[r, no head] & \widetilde{B}_{\Gamma}
\end{tikzcd}

\begin{tikzcd}
\pi \arrow[d, "\varphi"'] & \hat{\hat{\mathcal{E}}}(\pi, I') \arrow[d, "\wr\; \Psi_{I'}"] & 1 \arrow[r] & \pi \arrow[r] \arrow[d, "f_{\pi}"] & \hat{\hat{\mathfrak{S}}}(\pi, I') \arrow[r] \arrow[d, "f_{\hat{\hat{\mathfrak{S}}}}"] & {} \\
\pi' & \hat{\hat{\mathfrak{S}}}(\pi') \arrow[d, "p"] & 1 \arrow[r] & \pi' \arrow[r] & \hat{\hat{\mathcal{E}}}(\pi') \arrow[r] & {} \\
 & \hat{\hat{\mathcal{C}}}(\pi') & & & &
\end{tikzcd}