\begin{tikzcd}[column sep=small, row sep=large, nodes={font=\scriptsize}]
\mathrm{C}(I) \simeq \mathrm{C}_{\mathrm{aff}}(I_0) \arrow[dr, "\alpha"'] & & \\
& \mathrm{C}_{\mathrm{aff}}(n) \simeq \mathrm{C}_{\mathrm{aff}}(I_0)/\mathfrak{S}_{I_0} \simeq \mathrm{C}_{\cdots}(I)/\mathfrak{S}_{I_0} \arrow[dl, "\beta"'] \arrow[dr, "\beta'"] & \\
\mathrm{C}(n+1) \simeq \mathrm{C}(I)/\mathfrak{S}_I & & \mathrm{C}(n)
\end{tikzcd}

\begin{tikzcd}
S \arrow[rr, "\sim"] \arrow[dr, "\mathrm{can}"'] & & \widetilde{I}_0(\xi) \arrow[dl, "\mathrm{can}"] \\
& I_0 &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
\mathbb{D}_{I_0} \subset \mathfrak{S}_{I_0} \ [\subset \mathfrak{S}_I] & \\
\mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0) \subset \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0) \arrow[d, "\alpha"] \arrow[dr, "\alpha^{!}"'] & \\
\mathrm{C}_{\mathrm{aff}}(n) & \mathrm{C}_{\mathrm{aff}}(n) \simeq \mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0) / \mathbb{D}_{I_0}
\end{tikzcd}

\begin{tikzcd}
\widetilde{\mathrm{C}}_{\mathrm{st}}(S) \arrow[d, "\deg 2"] \arrow[dd, bend left=60, "\widetilde{\alpha}"] \\
\mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0) \arrow[d, "\alpha^{!}"] \\
\mathrm{C}_{\mathrm{aff}}(n) \simeq \widetilde{\mathrm{C}}_{\mathrm{st}}(S)/\mathbb{D}_S \simeq \mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0)/\mathbb{D}_{I_0}
\end{tikzcd}

\begin{tikzcd}
Z \arrow[d] & Z' \arrow[l] \arrow[d] \\
\widetilde{Z} & X' \arrow[l]
\end{tikzcd}

\begin{tikzcd}
Z \arrow[d] & Z' \arrow[l] \arrow[d] \\
X & X' \arrow[l]
\end{tikzcd}

\begin{tikzcd}
Z \arrow[d] & Z' \arrow[l] \arrow[d] \\
\widetilde{Z} \arrow[d] & \widetilde{Z}' \arrow[l] \arrow[d] \\
X & X' \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small]
\alpha'_i & & \alpha''_i \\
& 0 \arrow[ul, no head] \arrow[ur, no head] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\text{oct}^{+} \arrow[rr, "\approx"] \arrow[dr, "\approx"] \arrow[dd] & & \text{oct.\ gauches} \arrow[dr, "\approx"] \arrow[dd, no head, "=" description] & \\
& \text{gr.\ oct}^{+} \arrow[rr, "\approx"] \arrow[dd] & & \text{gr.\ oct.\ gauches}^{+} \arrow[dd] \\
\text{oct} \arrow[rr] \arrow[dr, "\approx"] & & \text{oct.\ gauches} \arrow[dr] & \\
& \text{gr.\ oct} \arrow[rr, "\approx"] & & \text{gr.\ oct.\ gauches}
\end{tikzcd}

\begin{tikzcd}[column sep=small]
& \vec{J} \arrow[dl] \arrow[dr] & \\
\vec{\mathfrak{Z}}_s \arrow[rr, dashed, "\psi_{\vec{a}}"] & & \vec{\mathfrak{Z}}_{s'}
\end{tikzcd}

\begin{tikzcd}[column sep=small]
\Omega(s) \arrow[rr, "\sim"] \arrow[dr, "\sim"'] & & \Omega(s') \\
& \Omega(D) \arrow[ur] &
\end{tikzcd}