\begin{tikzcd}[row sep=small]
  (E, x) \arrow[d, "f"'] & & \\
  (E', x') & (f(x) = x') & \Longleftrightarrow \; H \subset H'
\end{tikzcd}

\begin{tikzcd}[row sep=small]
  \partial X \arrow[r] \arrow[d, hook] & S \\
  X &
\end{tikzcd}

\begin{tikzcd}
  1 \arrow[r] & Z \arrow[r, "i"] & \operatorname{Aut}(Z, T) \arrow[r, "p"] &
  \operatorname{Aut}(Z) \arrow[r] \arrow[l, dashed, bend left=30, "q_t"] & 1
\end{tikzcd}

\begin{tikzcd}
  1 \arrow[r] & Z \arrow[r, "i"] & \operatorname{Aut}^{\pm 1}(T, Z) \arrow[r, "p"] &
  \{\pm 1\} \arrow[r] \arrow[l, bend left=30, "q"] & 1
\end{tikzcd}

\begin{tikzcd}
  P \arrow[r, dashed, "p'"] \arrow[d, dashed, "q'"'] & F \arrow[d, "q"] \\
  E \arrow[r, "p"] & S
\end{tikzcd}

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

\begin{tikzcd}[column sep=large]
  \underline{\mathrm{Hom}}(\underline{\mathrm{Tors}}(G), \mathcal{E})
  \arrow[r, "\approx"] \arrow[d, "\approx"'] &
  G\text{-}(\mathcal{E}) \arrow[dl, "{\Phi,\ \approx}" description] \\
  \underline{\mathrm{Hom}}(\mathcal{C}_E, \mathcal{E}) &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
  G = \mathrm{Gl}(n, \mathbf{R}) \arrow[rr, "s"] \arrow[dr, "\det"'] & & \{\pm 1\} \\
  & \mathbf{R}^* \arrow[ur, "\operatorname{sgn}"'] &
\end{tikzcd}

\begin{tikzcd}
  \mathrm{O}(n, k) \arrow[r, hook, "\mathrm{incl}"] \arrow[rr, bend right=20, "\det"'] &
  \mathrm{Gl}(n, k) \arrow[r, "\operatorname{sgn} \det"] & \{\pm 1\}
\end{tikzcd}

\begin{tikzcd}
  \mathfrak{S}_n \arrow[r, hook, "\mathrm{incl.}"] \arrow[rr, bend right=20, "\operatorname{sgn}"'] &
  \mathrm{O}(n, k) \arrow[r, "\det"] & \{\pm 1\}
\end{tikzcd}