\begin{tikzcd}[column sep=small]
0 \arrow[r] & I \arrow[r] \arrow[d, no head, "\mathrm{id}"'] & \bigotimes^E_A B \arrow[r] \arrow[d, "\sigma_{B/A}"] & \bigotimes^E_{A_0} B_0 \arrow[r] \arrow[d, "\sigma_{B_0/A_0}"] & 0 \\
0 \arrow[r] & I \arrow[r] & \bigotimes^F_A B \arrow[r] & \bigotimes^F_{A_0} B_0 \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & E' \arrow[r, bend left=8] \arrow[r, bend right=8] \arrow[dd, no head] & E'' \arrow[r, bend left=12] \arrow[r] \arrow[r, bend right=12] & E''' \\
E_0 \arrow[r, dashed] \arrow[dd, dashed, no head] & E'_0 \arrow[r, bend left=8] \arrow[r, bend right=8] & E''_0 \arrow[r, bend left=12] \arrow[r] \arrow[r, bend right=12] & E'''_0 \\
A \arrow[r] & A' \arrow[r, bend left=8] \arrow[r, bend right=8] & A'' \arrow[r, bend left=12] \arrow[r] \arrow[r, bend right=12] & A''' \\
A_0 \arrow[r] & A'_0 \arrow[r, bend left=8] \arrow[r, bend right=8] & A''_0 \arrow[r, bend left=12] \arrow[r] \arrow[r, bend right=12] & A'''_0
\end{tikzcd}

\begin{tikzcd}[column sep=small]
0 \arrow[r] & I \arrow[r] \arrow[d, Rightarrow, no head] & \bigotimes^n_A B \arrow[r, "\mathrm{pr}"] \arrow[d] & \bigotimes^n_{A_0} B_0 \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & I \arrow[r] & B \arrow[r] & B_0 \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
A \arrow[r] \arrow[d] & A' \arrow[r, bend left=8] \arrow[r, bend right=8] \arrow[d] & A'' \arrow[d] \\
A_0 \arrow[r] & A'_0 \arrow[r, bend left=8] \arrow[r, bend right=8] & A''_0
\end{tikzcd}

\begin{tikzcd}
B^{*} \arrow[r] \arrow[d, "i"] & C_0^{*} \arrow[r] \arrow[d, "j"] & H^1(I C^{\cdot}(A'/A)) \arrow[d, "\mathrm{id}"] \\
B \arrow[r] & C_0 \arrow[r] & H^1(I C^{\cdot}(A'/A))
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
e \arrow[r] & C^{\cdot}(A'/A, \underline{\mathrm{Hom}}(E, IE)) \arrow[r] \arrow[d] & C^{\cdot}(A'/A, \underline{\mathrm{Aut}}(E)) \arrow[r] \arrow[d] & C^{\cdot}(A'_0/A_0, \underline{\mathrm{Aut}}(E)) \arrow[r] & e \\
0 \arrow[r] & C^{\cdot}(A'/A, \underline{\mathrm{Hom}}(E, IE)) \arrow[r] & C^{\cdot}(A'/A, \underline{\mathrm{End}}(E)) \arrow[r] & C^{\cdot}(A'_0/A_0, \underline{\mathrm{End}}(E)) \arrow[r] & 0
\end{tikzcd}