\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \check{\ell}^{\,G^*}_\bullet = \ell \arrow[dl, "(1)"'] & \\
 \ell^{G}_\bullet[-1] \arrow[rr] & & \Delta^*(G) \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & {[\omega \xrightarrow{C} \omega^{(p)}]} \arrow[dl, "{(\mathrm{id},0)}"'] & \\
 {[\omega^{(p)} \xrightarrow{0} \omega]} \arrow[rr, "{(\mathrm{id},C)}"'] & & {[\omega^{(p)} \xrightarrow{0} \omega^{(p)}]} \arrow[ul, "{(0,\mathrm{id})}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \ell^{G(p)}_\bullet[1] \arrow[dl, "(1)"'] & \\
 \check{\ell}^{\,G^*(p)}_\bullet \arrow[rr] & & \Delta^*(G) \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & {[\omega^{(p^2)} \xrightarrow{0} \omega^{(p)}]} \arrow[dl, "{(\mathrm{id},0)}"'] & \\
 {[\omega^{(p)} \xrightarrow{C^{(p)}} \omega^{(p^2)}]} \arrow[rr, "{(\mathrm{id},0)}"'] & & {[\omega^{(p)} \xrightarrow{0} \omega^{(p)}]} \arrow[ul, "{(C,\mathrm{id})}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & {[\mathcal{M}^{(p)} \xrightarrow{V} \mathcal{M}^{(p)}]} \arrow[dl, "{(\mathrm{id},0)}"'] & \\
 {[\mathcal{M}^{(p)} \xrightarrow{F} \mathcal{M}^{(p)}]} \arrow[rr, "{(\mathrm{id},0)}"'] & & {\Delta^*(G) = [\mathcal{M} \xrightarrow{0} \mathcal{M}]} \arrow[ul, "{(V,\mathrm{id})}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & {[\mathcal{M} \xrightarrow{V} \mathcal{M}^{(p)}]} \arrow[dl, "{\partial_{\mathrm{I}},\ 1}"'] & \\
 {[\mathcal{M}^{(p)} \xrightarrow{F} \mathcal{M}]} \arrow[rr, "i_{\mathrm{I}}"'] & & \Delta^{*(p)} \arrow[ul, "\pi_{\mathrm{I}}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & {[\mathcal{M}^{(p)} \xrightarrow{F} \mathcal{M}]} \arrow[dl, "{\partial_{\mathrm{II}},\ 1}"'] & \\
 {[\mathcal{M} \xrightarrow{V} \mathcal{M}^{(p)}]} \arrow[rr, "i_{\mathrm{II}}"'] & & \Delta^* \arrow[ul, "\pi_{\mathrm{II}}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \check{\ell}'_\bullet \arrow[dl, "\partial_0"'] & \\
 \ell_\bullet[-1] \arrow[rr, "i_0"'] & & \Delta^* \arrow[ul, "\pi_0"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \varphi^*(\ell^{G^*}_\bullet)^*[-1] \arrow[dl, "\partial"'] & \\
 \varphi^*(\check{\ell}^{\,G^*}_\bullet) \arrow[rr] & & \Delta^*_{\mathrm{cris}}(G) \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & (\ell^{G}_\bullet)^{(p)}[-1] \arrow[dl, "\varepsilon^*(\partial)"'] & \\
 (\check{\ell}^{\,G^*}_\bullet)^{(p)} \arrow[rr] & & \Delta^*(G) \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\Delta^{*p} \arrow[rr, "F_{\Delta^*}"] \arrow[dr, "\pi_0^{(p)}"'] & & \Delta^* \\
 & \check\ell'^{(p)}_\bullet \arrow[ur, "\varepsilon^*(i_{\mathrm{II}})"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\Delta^* \arrow[rr, "V_{\Delta^*}"] \arrow[dr, "\varepsilon^*(\pi_{\mathrm{II}})"'] & & \Delta^{*(p)} \\
 & \ell^{(p)}_\bullet[-1] \arrow[ur, "{i_0^{(p)}}"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\ell_\bullet[-1] \arrow[r, "i_0"] \arrow[d, "{V_{\ell_\bullet}[-1]}"'] & \Delta^* \arrow[d, "V_{\Delta^*}"] \arrow[dr, "\varepsilon^*(\pi_{\mathrm{II}})"] & \\
\ell_\bullet[-1]^{(p)} \arrow[r, "i_0^{(p)}"'] & \Delta^{*(p)} & \ell^{(p)}_\bullet[-1] \arrow[l, "i_0^{(p)}"]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\check\ell'^{(p)}_\bullet \arrow[r, "{\pi_0\,\varepsilon^*(i_{\mathrm{II}})}"] \arrow[d, "\partial_0^{(p)}"'] & \check\ell'_\bullet \arrow[d, "\partial_0"] \\
\ell_\bullet[-1]^{(p)} \arrow[r, "0"'] & \ell_\bullet[-1]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathcal{M} \arrow[r, "V"] \arrow[d, "V"'] & \mathcal{M}^{(p)} \arrow[d, "V^{(p)}"] \\
\mathcal{M}^{(p)} \arrow[r, "V^{(p)}"'] & \mathcal{M}^{(p^2)}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & B \arrow[dl, "\partial_{\mathrm{I}}"'] & \\
 A \arrow[rr, "i_{\mathrm{I}}"'] & & \mathfrak{X}^p \arrow[ul, "\pi_{\mathrm{I}}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & A \arrow[dl, "\partial_{\mathrm{II}}"'] & \\
 B \arrow[rr, "i_{\mathrm{II}}"'] & & \mathfrak{X} \arrow[ul, "\pi_{\mathrm{II}}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & A^p \arrow[dl, "\partial_{\mathrm{II}}^{(p)}"'] & \\
 B^p \arrow[rr, "i_{\mathrm{II}}^{(p)}"'] & & \mathfrak{X}^p \arrow[ul, "\pi_{\mathrm{II}}^{(p)}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \varphi^*(b) \arrow[dl, "{\partial_{\mathrm{I}} = \varphi^*(\partial_0)}"'] & \\
 \varphi^*(a) \arrow[rr, "{i_{\mathrm{I}} = \varphi^*(i_0)}"'] & & \mathfrak{X}^p \arrow[ul, "{\pi_{\mathrm{I}} = \varphi^*(\pi_0)}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & b \arrow[dl, "\partial_0"'] & \\
 a \arrow[rr, "i_0"'] & & \mathfrak{X} \arrow[ul, "\pi_0"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & b^p \arrow[dl, "\partial_0^{(p)}"'] & \\
 a^p \arrow[rr, "i_0^{(p)}"'] & & X^p \arrow[ul, "\pi_0^{(p)}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & a^p \arrow[dl, "\varepsilon^*(\partial_{\mathrm{II}})"'] & \\
 b^p \arrow[rr, "\varepsilon^*(i_{\mathrm{II}})"'] & & X \arrow[ul, "\varepsilon^*(\pi_{\mathrm{II}})"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \mathcal{M}\overset{L}{\otimes}/V \arrow[dl, "\partial_{\mathrm{I}}"'] & \\
 \mathcal{M}\overset{L}{\otimes}/F \arrow[rr, "i_{\mathrm{I}}"'] & & \mathcal{M}^{(p)}\overset{L}{\otimes}/p \arrow[ul, "\pi_{\mathrm{I}}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \mathcal{M}\overset{L}{\otimes}/F \arrow[dl] & \\
 \mathcal{M}\overset{L}{\otimes}/V \arrow[rr] & & \mathcal{M}\overset{L}{\otimes}/p = \Delta^* \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \check\ell'_\bullet \arrow[dl, "\partial_0"'] & \\
 \ell_\bullet[-1] \arrow[rr, "i_0"'] & & \Delta^* \arrow[ul, "\pi_0"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \bar{\check\ell}'_\bullet \arrow[dl, "\bar\partial_0"'] & \\
 \bar\ell_\bullet[-1] \arrow[rr, "\bar i_0"'] & & \Delta^*(S') \arrow[ul, "\pi_0"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
M \arrow[r, two heads] \arrow[d, "V_M"'] \arrow[dr, "\mathrm{\acute{e}pi}", two heads] & \omega \arrow[d, "V_\omega"] \\
M^p \arrow[r, two heads] & \omega^{(p)}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
M^p \arrow[r, two heads] & t' \\
M^{(p)} \arrow[u] \arrow[r] & t'^{(p)} \arrow[u, "F_t"'] \arrow[ul, hook]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & t^p \arrow[d] \arrow[dr, "F_{t'}"] & \\
0 \arrow[r] & M \arrow[r] \arrow[d] & t' \arrow[r] & 0 \\
\omega \arrow[ur] \arrow[r, "V_\omega"'] & \omega^p \arrow[d] & & \\
 & 0 & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & L^{\prime}(\mathcal{M}) \arrow[dl] & \\
 L_\bullet(\mathcal{M}) \arrow[rr, no head] & & \Delta^*(\mathcal{M})^{(p)} \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & L_\bullet(\mathcal{M}) \arrow[dl] & \\
 L'(\mathcal{M}) \arrow[rr, no head] & & \Delta^*(\mathcal{M}) \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \mathcal{T}_1 \arrow[dl] & \\
 \mathcal{T}_0 \arrow[rr, no head] & & \mathcal{T}_1 \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\varphi^*(\mathcal{T}_1) \arrow[r, "\alpha_1"] \arrow[dr, "\varphi^*(\beta)"'] & \Delta^*(\mathcal{M})^{(p)} \\
 & \varphi^*\varepsilon^*(\Delta^*(\mathcal{M})) \simeq \Delta^*(\mathcal{M})^{(p)} \arrow[u, Rightarrow, "\mathrm{id}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & {\mathcal{T}_1^{(\omega)} = [\omega \xrightarrow{V_\omega} \omega^{p}]} \arrow[dl, "1"'] & \\
 {\mathcal{T}_0^{(\omega)} = [\omega^{V} \xrightarrow{F_\omega} \omega]} \arrow[rr] & & {\mathcal{T}_1^{(\omega)} = [\omega^{(p)} \xrightarrow{0} \omega^{(p)}]} \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathcal{T}_1(\omega) \arrow[r, "v_1"] \arrow[d, Rightarrow, "\beta_\omega"'] & \mathcal{T}_1(\omega') \arrow[d, Rightarrow, "\beta_{\omega'}"] \\
\varepsilon^*\Delta^*(\varphi^*\omega) \arrow[d, no head, "\|" description] & \varepsilon^*(\Delta^*(\varphi^*\omega')) \arrow[d, no head, "\|" description] \\
\Delta^*(\omega) \arrow[r, "\varepsilon^*(u)"'] & \Delta^*(\omega')
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\varphi^*(\mathcal{T}(\omega)) \arrow[r, no head, "\alpha_\omega \,\sim"] \arrow[d, "\varphi^*(v)"'] & \mathcal{T}(\varphi^*\omega) \arrow[d, "\mathcal{T}(u)"] \\
\varphi^*(\mathcal{T}(\omega')) \arrow[r, no head, "\alpha_{\omega'} \,\sim"'] & \mathcal{T}(\varphi^*\omega')
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & L' \arrow[dl] & \\
 L_\bullet \arrow[rr] & & \Delta(\mathcal{M}) \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \mathcal{M}_{/V} \arrow[dl] & \\
 \mathcal{M}_{/F} \arrow[rr] & & {\mathcal{M}_{/p}^{(\varphi)} = \varphi^*(\Delta(\mathcal{M}))} \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
J \arrow[r, hook] \arrow[d] & B \arrow[r] \arrow[d, "\mu"] & A \arrow[d] \arrow[dr, "{\lambda \mapsto \lambda^p}"] & \\
JB' \arrow[r, hook] & B' \arrow[r] & B'/JB' \arrow[r] \arrow[ddr] & A \\
J^{(p)} & B \arrow[d, dashed] & B/J^{(p)}B \arrow[d] & \\
 & {} & B \arrow[r] & C \arrow[uu]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
J^{(p)}B \arrow[dr] & J \arrow[r, hook] \arrow[dr] & B \arrow[r, no head] \arrow[dr] & A \\
 & 0 & I \arrow[r, hook] & C' \arrow[r] & A
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & \Delta^*(\mathcal{M}) \arrow[dl, no head] \arrow[dr, no head] & \\
 & \tilde{D}_1 & & \tilde{D}_0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & t^*[1] \arrow[dl, "1"'] & \\
 \omega^* \arrow[rr] & & \tilde{\Delta}^*(\mathcal{M}) \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \check{\ell}'_\bullet[1] \arrow[dl] & \\
\ell \arrow[rr] & & \Delta \arrow[ul]
\end{tikzcd}

\begin{tikzcd}
M \arrow[r, "\varphi"] \arrow[d, no head] &
P' \otimes \mathcal{O}_S + Q' \otimes \mathcal{O}_S \arrow[d, "u"] \\
M \arrow[r, "\varphi'"] &
P' \otimes \mathcal{O}_S + Q' \otimes \mathcal{O}_S
\end{tikzcd}

\begin{tikzcd}
F_S^{*}(M) \arrow[r, "F_S^{*}(\varphi)"] \arrow[d, "F_M"'] &
P' \otimes \mathcal{O}_S + Q' \otimes \mathcal{O}_S \arrow[d, "F_{M,\varphi}"] \\
M \arrow[r, "\varphi"] &
P' \otimes \mathcal{O}_S + Q' \otimes \mathcal{O}_S
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
S_0 \arrow[r, no head] \arrow[d, no head] & S \arrow[d, no head] \\
k \arrow[r, no head] & W
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
S \arrow[r, "F_S"] \arrow[d, no head] & S \arrow[d, no head] \\
\operatorname{Spf} W \arrow[r, "F_W"] \arrow[dr, no head] &
\operatorname{Spf}(W) \arrow[d, no head] \\
 & \operatorname{Spf} \mathbb{Z}_p
\end{tikzcd}

\begin{tikzcd}
F_S^{*}(E_S) \arrow[r, "F_M"] & E_S \\
E_S \arrow[u, no head, "\wr"] \arrow[ur, "\Phi"'] &
\end{tikzcd}

\begin{tikzcd}
E_S \arrow[r, "u"] & E'_S \\
E_S \arrow[u, "F_M"] \arrow[r, dashed, "F_S^{*}(u) = u^{\sigma}"'] &
E'_S \arrow[u, "F_{M'}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\mathbb{Z}_p \arrow[r, no head] \arrow[d, no head] & A \arrow[d, no head] \\
\mathbb{Q}_p \arrow[r, no head] & K
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
A \arrow[r, no head] \arrow[d, no head] &
A \otimes_{\mathbb{Z}_p} W \arrow[d, no head] \\
\mathbb{Z}_p \arrow[r, no head] & W
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
A \arrow[r, no head] & A \otimes_{\mathbb{Z}_p} W \arrow[r, no head] & A \otimes_{\mathbb{Z}_p} A \\
\mathbb{Z}_p \arrow[u, no head] \arrow[r, no head] & W \arrow[u, no head] \arrow[r, no head] & A \arrow[u, no head]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
T_p \simeq A \arrow[r, no head] & T_p \otimes A \simeq A \otimes_{\mathbb{Z}_p} A \arrow[r, no head] & T_p \otimes C \simeq A \otimes_{\mathbb{Z}_p} C \\
\mathbb{Z}_p \arrow[u, no head] \arrow[r, no head] & A \arrow[u, no head] \arrow[r, no head] & C \arrow[u, no head]
\end{tikzcd}

\begin{tikzcd}[column sep=small]
0 \arrow[r] & \mathfrak{t} \otimes C(1) \arrow[r] & T_p \otimes C \arrow[r] \arrow[l, dashed, bend left=30] & \check{\mathfrak{t}}^{*} \otimes C \arrow[r] \arrow[l, dashed, bend left=30] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
W \arrow[r, hook] \arrow[d, hook] & V \arrow[d, hook] \\
K \arrow[r, hook] & L
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\pi = \operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \arrow[r] & G(\mathbb{Q}_p) \\
\pi(p) = \operatorname{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p) \arrow[u] &
\end{tikzcd}