\begin{tikzcd}
1 \arrow[r] & \displaystyle\varprojlim_{n \text{ premier à } p} \mu_n(\overline{K}) \arrow[r] & \pi_1(K) \arrow[r] & \pi_1(k) \arrow[r] \arrow[d, no head, "=" description] & 1 \\
 & & & \pi_1(V) &
\end{tikzcd}

\begin{tikzcd}
\mu_n(S) \arrow[r] & \mathbb{G}_m(S) \arrow[r] & \mathbb{G}_m(S) \arrow[r] & \mathrm{Rev}(S, \mu_n) \arrow[d] \\
 & & & \mathrm{Pic}(S) \arrow[d, "\otimes n"] \\
 & & & \mathrm{Pic}(S)
\end{tikzcd}

\begin{tikzcd}
 & (E_{\pi} \times X)/\pi \arrow[dl] \arrow[dr] & \\
B_{\pi} = E_{\pi}/\pi & & X/\pi
\end{tikzcd}

\begin{tikzcd}
\textbf{Équiv}\bigl(\mathrm{Ens}(\pi), \mathrm{Ens}(\pi')\bigr) \arrow[d, "u \,\mapsto\, u^{[-1]}"'] & \approx & \mathrm{Bit}(\pi', \pi) \arrow[d] \\
\textbf{Équiv}\bigl(\mathrm{Ens}(\pi'), \mathrm{Ens}(\pi)\bigr) \arrow[rr, "\approx"] & & \mathrm{Bit}(\pi, \pi')
\end{tikzcd}

\begin{tikzcd}
\pi_0\bigl(\mathbf{Bit.pt}(\pi', \pi)\bigr) \arrow[d] & \simeq & \mathrm{Isom}(\pi', \pi) \arrow[d, "\text{surj.}"] \\
\pi_0\bigl(\mathbf{Bit}(\pi', \pi)\bigr) & \simeq & \mathrm{Isom}(\pi', \pi)/\mathrm{Int}(\pi) = \mathrm{Isomext}(\pi', \pi)
\end{tikzcd}