\begin{tikzcd}
R \arrow[r, "\mathrm{pr}_{1} \circ v"] \arrow[d, "\mathrm{pr}_{2} \circ v"'] & X \arrow[d, "p"] \\
X \arrow[r, "p"'] & X
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathcal{E} \arrow[r, "\rho"] \arrow[d] &
  \mathcal{A}(X) \otimes \mathrm{S}\mathcal{F} \arrow[d]
  \arrow[dl, "\mathcal{A}(\eta) \otimes \mathrm{id}"] \\
\mathrm{S}\mathcal{F} & \mathcal{F} \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=large]
G \times_{S} G \times_{S} \mathbb{V}(\mathcal{E})
  \arrow[r, "\mathrm{id}_{G} \times \theta"]
  \arrow[d, "\mu \times \mathrm{id}_{\mathbb{V}(\mathcal{E})}"'] &
G \times_{S} \mathbb{V}(\mathcal{E}) \arrow[d, "\theta"] \\
G \times_{S} \mathbb{V}(\mathcal{E}) \arrow[r, "\theta"'] &
\mathbb{V}(\mathcal{E})
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
S \times_{S} \mathbb{V}(\mathcal{E})
  \arrow[rr, "\varepsilon \times \mathrm{id}_{\mathbb{V}(\mathcal{E})}"] & &
G \times_{S} \mathbb{V}(\mathcal{E}) \arrow[dl, "\theta"] \\
 & \mathbb{V}(\mathcal{E}) \arrow[ul, "\simeq"] &
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathcal{A} \otimes \mathcal{A} \otimes \mathcal{A} &
\mathcal{A} \otimes \mathcal{E}
  \arrow[l, "\mathrm{id}_{\mathcal{A}} \otimes \rho"'] \\
\mathcal{A} \otimes \mathcal{E}
  \arrow[u, "\Delta \otimes \mathrm{id}_{\mathcal{E}}"] &
\mathcal{A} \arrow[l, "\rho"'] \arrow[u, "\rho"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\mathcal{O}_{S} \otimes \mathcal{E} & &
\mathcal{A} \otimes \mathcal{E}
  \arrow[ll, "\eta \otimes \mathrm{id}_{\mathcal{E}}"'] \\
 & \mathcal{E} \arrow[ul, "\simeq"] \arrow[ur, "\rho"'] &
\end{tikzcd}

\begin{tikzcd}[row sep=small, nodes={font=\scriptsize}]
\underline{G}(k) \arrow[r] \arrow[d] & \underline{G}(k') \arrow[d] \\
\underline{\Pi}(k') \times [\underline{G}_{m}]_{\underline{\mathbb{Q}}_{p}} \arrow[r] & \underline{\Pi}_{p}(k) \times [\underline{G}_{m}]_{\underline{\mathbb{Q}}_{p}}
\end{tikzcd}

\begin{tikzcd}
\underline{\mathbb{Q}} &
\bar{\underline{\mathbb{Q}}}^{*} \arrow[l, "u_{a}^{*}"'] &
\bar{\underline{\mathbb{Q}}}^{*} \arrow[l, "\lambda^{a}"']
  \arrow[ll, bend left=25, "|\lambda|"']
\end{tikzcd}

\begin{tikzcd}
M \arrow[dr] & A \otimes_{k} \overline{M} \arrow[l, "\approx"'] \arrow[d] \\
 & \overline{M}
\end{tikzcd}

\begin{tikzcd}[column sep=large]
A^{G'} \arrow[r, "T^{n}_{G'}"] \arrow[d, "A^{u}"'] &
  (A^{G'})^{\mathfrak{S}_{n}} \arrow[d, "A^{u \times 1_{\mathfrak{S}_{n}}}"] \\
A^{G} \arrow[r, "T^{n}_{G}"'] & (A^{G})^{\mathfrak{S}_{n}}
\end{tikzcd}

\begin{tikzcd}[column sep=large]
A(G') \arrow[r, "T_{G'}"] \arrow[d, "A(u)"'] &
  1 + \widehat{A(G')^{\bullet}}^{\,+} \arrow[d, "\widehat{A(u)}"] \\
A(G) \arrow[r, "T_{G}"'] & 1 + \widehat{A(G)^{\bullet}}^{\,+}
\end{tikzcd}

\begin{tikzcd}
E_{T} \arrow[d] \\ B_{T}
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathbb{Z}(\hat{T}) \arrow[r] & H^{*}(B_{T}, \mathbb{Z}) \\
A(T) \arrow[u, "\mathrm{alg}"] \arrow[ur, "\text{top.\ via Chern}"']
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathbb{Z}[\hat{T}] \arrow[r, "\text{hom.\ car.}"] & H^{*}(X, \mathbb{Z}) \\
A(T) \arrow[u] \arrow[ur, "\text{top.\ via Chern}"']
\end{tikzcd}

\begin{tikzcd}[column sep=large]
K(G) \arrow[r, "\text{ass.}"] \arrow[d, "c_{G}"'] & K(X) \arrow[d, "c_{X}"] \\
\widetilde{CK}(G) \arrow[r, "\text{hom.\ car.}"'] &
  \widetilde{CK}(X) = \widetilde{A}(X)
\end{tikzcd}

\begin{tikzcd}[column sep=large]
CK(G') \arrow[r, "\varphi^{*}"] \arrow[dr, "\tau_{P'}"'] & CK(G) \arrow[d, "\tau_{P}"] \\
 & A(X)
\end{tikzcd}

\begin{tikzcd}
A_{i} \arrow[r, "\varphi"] \arrow[d, "u_{i}"'] & A_{j} \arrow[d, "u_{j}"] \\
A_{i} \arrow[r, "\varphi"'] & A_{j}
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathcal{C}^{\underline{\mathcal{U}}} \arrow[r]
  \arrow[dr, "I^{\underline{\mathcal{U}}}"'] &
  \mathcal{C}^{\underline{V}} \arrow[d, "I^{\underline{V}}"] \\
 & \mathcal{C}^{k}
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathcal{C}^{\underline{\mathcal{U}}} \arrow[r, "F^{\vee}"] \arrow[d] &
  \mathcal{C}^{\underline{\mathcal{U}}} \arrow[d] \\
\mathcal{C}^{k} \arrow[r, "F^{\vee}"'] & \mathcal{C}^{k}
\end{tikzcd}

\begin{tikzcd}[column sep=large]
k^{\underline{\mathcal{U}}} \times k^{\underline{\mathcal{U}}}
  \arrow[r, "T"] \arrow[d, "I^{\underline{\mathcal{U}}} \otimes I^{\underline{\mathcal{U}}}"'] &
  k^{\underline{\mathcal{U}}} \arrow[d, "I^{\underline{\mathcal{U}}}"] \\
k^{k} \times k^{k} \arrow[r, "T"'] & k^{k}
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\underline{\mathcal{U}} \arrow[r, "\Delta"] \arrow[d, "\vee"'] &
  \underline{\mathcal{U}} \otimes \underline{\mathcal{U}} \arrow[d, "\vee \otimes \vee"] \\
\underline{\mathcal{U}} \arrow[r, "\Delta"'] &
  \underline{\mathcal{U}} \otimes \underline{\mathcal{U}}
\end{tikzcd}