\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  \text{Formes bilinéaires } E \times E \to \mathcal{O}_S \arrow[rr, "\sim"] \arrow[dr, "\sim"'] & & \text{structures d'algèbre alternée sur } E' \\
  & \mathrm{Hom}_{\mathcal{O}_S}(E, E') \arrow[ur, "\sim"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
    0 \arrow[r] & \det V \arrow[r] & E \arrow[r] & V \arrow[r] & 0 \\
    0 & \mathcal{O}_S \arrow[l] & E' \arrow[l] & \check{V}' \arrow[l] & 0 \arrow[l]
\end{tikzcd}

\begin{tikzcd}
  \widetilde{G} \arrow[r, "F"] \arrow[dr, "\mathrm{ad}_{\widetilde{G}}"'] & G \arrow[d, "\rho_{\widetilde{G}}"] \arrow[dr, "\mathrm{ad}_G"] & \\
  & \underline{\mathrm{Aut}}_{S\text{-}\mathrm{gr}}(\widetilde{G}) \arrow[r] & \underline{\mathrm{Aut}}_{S\text{-}\mathrm{gr}}(G)
\end{tikzcd}

\begin{tikzcd}
  \widetilde{G} \times \widetilde{G} \arrow[r, "\mathrm{ad}_{\widetilde{G}}"] \arrow[d, "F \times \mathrm{id}_{\widetilde{G}}"] & \widetilde{G} \arrow[d, "\mathrm{id}"] \\
  G \times \widetilde{G} \arrow[r, "\rho"] \arrow[d, "\mathrm{id}_G \times F"] & \widetilde{G} \arrow[d, "F"] \\
  G \times G \arrow[r, "\mathrm{ad}_G"] & G
\end{tikzcd}

\begin{tikzcd}
  \widetilde{\mathfrak{g}} \times \widetilde{\mathfrak{g}} \arrow[r, "\mathrm{ad}_{\widetilde{\mathfrak{g}}}"] \arrow[d, "\mathfrak{f} \times \widetilde{\mathfrak{g}}"] & \widetilde{\mathfrak{g}} \arrow[d, "\mathrm{id}"] \\
  \mathfrak{g} \times \widetilde{\mathfrak{g}} \arrow[r] \arrow[d] & \widetilde{\mathfrak{g}} \arrow[d, "\mathfrak{f}_{\widetilde{\mathfrak{g}}}"] \\
  \mathfrak{g} \times \mathfrak{g} \arrow[r, "\mathrm{ad}_{\mathfrak{g}}"] & \mathfrak{g}
\end{tikzcd}