\begin{tikzcd}
  \widetilde{\partial X} \arrow[d, "p"] & \\
  \partial X \arrow[r, hook'] & X
\end{tikzcd}

\begin{tikzcd}
  \mathrm{Dr}(N) \arrow[d, "\text{appl.\ str.\ croissante}"] \\
  \tfrac{1}{2}\mathrm{simpl}
\end{tikzcd}

\begin{tikzcd}
  \Theta(D) \arrow[d] \\
  \mathrm{Dr}(N) \arrow[d] \\
  (\tfrac{1}{2}\mathrm{simpl})^*
\end{tikzcd}

\begin{tikzcd}
  \widetilde{X}_i \arrow[d, "p_i"'] & \supset & \partial\widetilde{X}_i
  \arrow[d, "q_{i-1}"] \\
  X_i & \supset & X_{i-1}
\end{tikzcd}

\begin{tikzcd}
  \mathcal{F}_n^\circ \arrow[r, "E_n^\circ"] \arrow[d, hook] & G_n\text{-ens}
  \arrow[d, "\text{oubli}"] \\
  \mathcal{F}_n \arrow[r, "E_n"] & G_{n-1}\text{-ens}
\end{tikzcd}

\begin{tikzcd}
  E_1(X_1) = \partial\widetilde{X}_1 \arrow[d] \\
  X_0
\end{tikzcd}

\begin{tikzcd}
  E_{n-1}(\partial\widetilde{X}_n) \arrow[d] \\
  E_{n-1}(X_{n-1})
\end{tikzcd}

\begin{tikzcd}
  \mathcal{F}_n^\circ \arrow[r, hook] \arrow[d, hook] &
  (G_n\text{-ens}) \arrow[d, "\psi_n"] \\
  \mathcal{F}_n \arrow[r] & (\mathrm{Casc})_n
\end{tikzcd}

\begin{tikzcd}
  \mathcal{F}_n^\circ \arrow[d, hook] & G_n\text{-ens} \arrow[l]
  \arrow[d, dashed] \\
  \mathcal{F}_n & (\mathrm{Casc})_n \arrow[l, "\psi_n"]
\end{tikzcd}