\begin{tikzcd}[column sep=small]
X \arrow[dr, "f"'] & & Y \arrow[dl, "g"] \\
& S &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
G \arrow[dr, "f_1"'] & & H \arrow[dl, "g_1"] \\
& \Sigma &
\end{tikzcd}

\begin{tikzcd}
e_\Sigma \arrow[r, "\sigma"] \arrow[d, "f_1(g)"'] & e_\Sigma \arrow[d, "g_1(h)"] \\
e_\Sigma \arrow[r, "\sigma'"'] & e_\Sigma
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\pi_2(Z) \arrow[r] & \pi_2(Y) \arrow[r] \arrow[d] & \pi_1(F) \arrow[r] \arrow[d, "\text{épi}"] & \pi_1(Z) \arrow[r] \arrow[d] & \pi_1(Y) \arrow[r] \arrow[d, "\wr"] & \pi_0(F) \arrow[r] \arrow[d, "\wr"] & \pi_0(Z) \arrow[r] \arrow[d] & 1 \\
& 1 \arrow[r] & \pi_1(\Phi) \arrow[r] & \pi_1(\widetilde{Z}) \arrow[r] & \pi_1(\widetilde{Y}) \arrow[r] & \pi_0(\Phi) \arrow[r] & \pi_0(\widetilde{Z}) \arrow[r] & 1
\end{tikzcd}

\begin{tikzcd}[row sep=small]
F \arrow[d, hook] & & \Phi \arrow[d, hook] & \\
X \arrow[d] & & \mathrm{Cat}\, G \arrow[r, "\approx"] \arrow[d] & \widetilde{X} \arrow[d] \\
S & & \mathrm{Cat}\, \Sigma \arrow[r, "\approx"] & \widetilde{S}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\pi_2(X) \arrow[r] & \pi_2(S) \arrow[r] & \pi_1(F) \arrow[r] & \pi_1(X) \arrow[r] \arrow[d, no head, "\wr"] & \pi_1(S) \arrow[r] \arrow[d, no head, "\wr"] & \pi_0(F) \arrow[d, no head, "\wr"] \\
& 1 \arrow[r] & \pi_1(\widetilde{F}) \arrow[r] & \pi_1(\widetilde{X}) \arrow[r] & \pi_1(\widetilde{S}) \arrow[r] & \pi_0(\Phi) \arrow[r] & 1
\end{tikzcd}

\begin{tikzcd}
\mathcal{D} \arrow[r, "\varphi"] & \mathcal{C}
\end{tikzcd}

\begin{tikzcd}
\mathcal{D} \arrow[r, "\varphi"] \arrow[d, "u"', "\wr"] & \mathcal{C} \arrow[d, "v"] \\
\mathcal{D}' \arrow[r, "\varphi'"'] & \mathcal{C}'
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathcal{D} \arrow[r, "\varphi"] \arrow[d, bend right=30, "u_1"'] \arrow[d, bend left=30, "u_2"] & \mathcal{C} \arrow[d, bend right=30, "v_1"'] \arrow[d, bend left=30, "v_2"] \\
\mathcal{D}' \arrow[r, "\varphi'"'] & \mathcal{C}'
\end{tikzcd}

\begin{tikzcd}
\varphi' u_1 \arrow[r, "\varphi' * \lambda", "\sim"'] \arrow[d, "\alpha_1"', "\wr"] & \varphi' u_2 \arrow[d, "\alpha_2", "\wr"'] \\
v_1 * \varphi \arrow[r, "\mu * \varphi"', "\sim"] & v_2 * \varphi
\end{tikzcd}

\begin{tikzcd}[row sep=small]
(E) & 1 \arrow[r] & N \arrow[r] & G \arrow[r] & H \arrow[r] & 1 \\
(E') & 1 \arrow[r] & N \arrow[r] \arrow[u, "\mathrm{id}"] & G' \arrow[r] \arrow[u] & H' \arrow[r] \arrow[u, "u"'] & 1
\end{tikzcd}

\begin{tikzcd}[row sep=small, column sep=small]
(S) & \pi_1 H \arrow[r] & \pi_0 N \arrow[r] & \pi_0 G \arrow[r] & \pi_0 H \arrow[r] & 1 \\
(S') & \pi_1 H' \arrow[r] \arrow[u] & \pi_0 N \arrow[r] \arrow[u, no head, "\parallel" description] & \pi_0 G' \arrow[r] \arrow[u] & \pi_0 H' \arrow[r] \arrow[u] & 1
\end{tikzcd}

\begin{tikzcd}[row sep=small]
H \arrow[r] & H/N^0 \\
H' \arrow[r] \arrow[u, "u"] & H'/N^0 \arrow[u, "\hat{u}"']
\end{tikzcd}

\begin{tikzcd}[row sep=small]
\pi_i H \arrow[r] & \pi_i(H/N^0) \\
\pi_i H' \arrow[r] \arrow[u, "\pi_i u"] & \pi_i(H'/N^0) \arrow[u, "\pi_i \hat{u}"']
\end{tikzcd}

\begin{tikzcd}[row sep=small]
\pi_1 H \arrow[r] & N \\
\pi_1 H' \arrow[r] \arrow[u, "\pi_1(u)"] & N \arrow[u, no head, "\parallel" description]
\end{tikzcd}

\begin{tikzcd}[row sep=small]
1 \arrow[r] & N \arrow[r] & \widetilde{G}_0 \arrow[r] & \mathfrak{G} \arrow[r] & 1 \\
& & & \pi \arrow[u] \arrow[ul, dashed, "\varphi"] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& \mathrm{SO} & & & \\
& & \mathrm{PLO} \arrow[ul] \arrow[dl] \arrow[drr] & & \\
& \mathrm{TUBO} \arrow[dl] \arrow[r] & \mathrm{SOB} \arrow[dr, "\partial"] & & \mathrm{SOB} \arrow[dl, "-\partial\ (\simeq \partial)"] \\
\mathrm{Kc} & & & \mathrm{CO} &
\end{tikzcd}

\begin{tikzcd}[row sep=small]
\pi_1(H) \arrow[r] \arrow[d, no head, "\parallel" description] & \pi_0(N) \arrow[d, no head, "\parallel" description] \\
\pi \arrow[r] \arrow[dr, "\varphi"'] & N \\
& \mathfrak{z}(N) \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\pi_1(H) \arrow[r] \arrow[d, no head, "\parallel" description] & N \arrow[r] & \pi_0(G) \arrow[r] & \pi_0(H) \arrow[r] \arrow[d, no head, "\parallel" description] & 1 \\
\pi \arrow[r, "\varphi"'] & \mathfrak{z}(N) \arrow[u, hook] & & \mathfrak{G} &
\end{tikzcd}

\begin{tikzcd}
  \pi_1(H) \arrow[r] \arrow[dd] & \pi_0 N \arrow[d, no head, "\parallel" description] \\
  & N \\
  \pi_0(\pi) = \pi \arrow[r, "\varphi"'] & \mathfrak{z}(N) \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}
  \pi_1 H \arrow[r] \arrow[d, "\partial"'] & N \arrow[r] & \pi_0 G \arrow[r] & \pi_0 H \arrow[r] & 1 \\
  \pi \arrow[r, "\varphi"] & \mathfrak{z}(N) \arrow[u, hook] & & &
\end{tikzcd}

\begin{tikzcd}
  \pi_1 H \arrow[r] \arrow[dd, "\partial"'] & G_1 \\
  & N \arrow[u, hook] \\
  \pi \arrow[r, "\varphi"] & \mathfrak{z}(N) \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}
  (b') & 1 \arrow[r] & \pi \arrow[r] \arrow[d, no head, "\parallel" description] & \widetilde{H}' \arrow[r] \arrow[d] & H' \arrow[r] \arrow[d, "u"] & 1 \\
  (b) & 1 \arrow[r] & \pi \arrow[r] & \widetilde{H} \arrow[r] & H \arrow[r] & 1
\end{tikzcd}

\begin{tikzcd}
  (a') & 1 \arrow[r] & \widetilde{H}'_0 \arrow[r] \arrow[d] & \widetilde{H}' \arrow[r] \arrow[d] & \mathfrak{G}' \arrow[r] \arrow[d, hook] & 1 \\
  (c) & 1 \arrow[r] & \widetilde{H}_0 \arrow[r] & \widetilde{H} \arrow[r] & \mathfrak{G} \arrow[r] & 1
\end{tikzcd}

\begin{tikzcd}
  1 \arrow[r] & N \arrow[r] \arrow[d, no head, "\parallel" description] & G' \arrow[r] \arrow[d] & H' \arrow[r] \arrow[d] & 1 \\
  1 \arrow[r] & N \arrow[r] & G \arrow[r] & H \arrow[r] & 1
\end{tikzcd}

\begin{tikzcd}
  \widetilde{H}'_0 \arrow[r] \arrow[d, dashed] & H' \arrow[d, dashed] \\
  \widetilde{H}_0 \arrow[r] & H
\end{tikzcd}