\begin{tikzcd}
  L'_{q+1} \arrow[r] \arrow[d] & L'_{q} \arrow[r] \arrow[d] & 0 \\
  0 \arrow[r] & M \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=normal, nodes={font=\scriptsize}]
  0 \arrow[r] & \Gamma^{n}\Phi_{*} \arrow[r] \arrow[d, "u_{0}"] & \Gamma^{n-1}\Phi_{*} \otimes \Psi_{*} \arrow[r] \arrow[d, "u_{1}"] & \Gamma^{n-2}\Phi_{*} \otimes \Lambda^{2}\Psi_{*} \arrow[r] \arrow[d, "u_{2}"] & \cdots \arrow[r] & \Lambda^{n}\Psi_{*} \arrow[r] \arrow[d, "u_{n}"] & 0 \arrow[d] \\
  0 \arrow[r] & \Phi_{*} \arrow[r, "\wedge T"'] & \Lambda^{2}\Phi_{*} \arrow[r, "\wedge T"'] & \Lambda^{3}\Phi_{*} & \cdots \arrow[r, "\wedge T"'] & \Lambda^{n+1}\Phi_{*} \arrow[r, "\wedge T"'] & \Lambda^{n+2}\Phi_{*}
\end{tikzcd}

\begin{tikzcd}
  \cdots \arrow[r] & k \arrow[r, "n"] \arrow[d, "u_{i}"] & k \arrow[r] \arrow[d, "u_{i-1}"] & \cdots \\
  \cdots \arrow[r] & L_{i} \arrow[r, "d_{i}"] & L_{i-1} \arrow[r] & \cdots
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & \Lambda^{i-1}\Psi_{*} \arrow[r] \arrow[d, "n\,\mathrm{id}"] & \Lambda^{i}\Phi_{*} \arrow[r] \arrow[d] & \Lambda^{i}\Psi_{*} \arrow[r] \arrow[d, no head, "\mathrm{id}"] & 0 \\
0 \arrow[r] & \Lambda^{i-1}\Psi_{*} \arrow[r] & \Phi_{\bullet}(i,n) \arrow[r] & \Lambda^{i}\Psi_{*} \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & \Psi^{i-1} \arrow[r, "\partial_i"] \arrow[d, "n\,\mathrm{id}"] & \Phi^{i} \arrow[r, "u_i"] \arrow[d] & \Psi^{i} \arrow[r] \arrow[d, "\mathrm{id}"] & 0 \\
0 \arrow[r] & \Psi^{i-1} \arrow[r] & \Phi(i,n) \arrow[r] & \Psi^{i} \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & \Psi^{i-1} \arrow[r] \arrow[d] & \Phi(i,-) \arrow[r] \arrow[d] & \Psi^{i} \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & \Psi^{j-1} \arrow[r] & \Phi(j,-) \arrow[r] & \Psi^{j} \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & \Psi^{i-1} \arrow[r] \arrow[d, "p\,\mathrm{id}"] & \Phi(i,n) \arrow[r] \arrow[d, "\alpha"] & \Psi^{i} \arrow[r] \arrow[d, "q\,\mathrm{id}"] & 0 \\
0 \arrow[r] & \Psi^{i-1} \arrow[r] & \Phi(i,m) \arrow[r] & \Psi^{i} \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \Psi^{i-1} \arrow[r] \arrow[d, "u\circ d"] & \Phi^{i} \arrow[r] \arrow[d] & \Psi^{i} \arrow[d, no head, "="] & \\
0 \arrow[r] & \Psi^{i-1} \arrow[r, "D"] \arrow[d, "\wr"] & \Phi(i,-) \arrow[r] \arrow[d, "\alpha=?"] & \Psi^{i} \arrow[r] \arrow[d, "\wr"] & 0 \\
0 \arrow[r] & \Psi^{i-1} \arrow[r] & \Phi(i,m) \arrow[r, "u"] & \Psi^{i} \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & \Psi^{i-1} \arrow[r] \arrow[d, "u\circ d"] & \Phi^{i}(k) \arrow[r] \arrow[d] & \Psi^{i} \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & \Psi^{i-1} \arrow[r, "i"] & \Phi(i,-) \arrow[r] \arrow[d, "\alpha=?"] & \Psi^{i} \arrow[r] & 0 \\
 & & \Psi^{i-1} & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & & & \Psi^{i} \arrow[d, "\alpha"'] \arrow[dr] & \\
0 \arrow[r] & \Psi^{i-1} \arrow[r] & \Phi(j,-) \arrow[r] & \Psi^{i} \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
C \arrow[r, "c_{ab}"] \arrow[dr, "c"'] \arrow[rr, bend left=30, "\rho_{ab}"] & \widehat{C}_{ab} \arrow[r, "\text{rest.}\,\rho_{ab}"] \arrow[d, "\text{oubli}\ \omega_C"] & \widehat{A}_{ab} \arrow[d, "\text{oubli}\ \omega_A"] \\
 & \widehat{C} \arrow[r, "\text{rest}\,\rho"] & \widehat{A}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=normal]
 & A \subset C \arrow[dl, "\mathrm{id}"'] \arrow[d, "\rho\ \wr"] \arrow[dr, "\mathrm{id}"] \arrow[drr, "\rho'\ \wr"] & & \\
A \subset \widehat{A}_{\mathcal{T}} \subset \widehat{A} & A \arrow[r, hook] & (\widehat{A^{\circ}}_{\mathcal{T}'})^{\circ} \arrow[r, hook] & (\widehat{A^{\circ}})^{\circ}
\end{tikzcd}

\begin{tikzcd}
\mathcal{C} \arrow[r, "\alpha_{\bullet}\ \approx"] \arrow[d, "(\alpha^{\bullet})^{\circ}\ \wr"'] & \mathbb{K}_{\bullet} \\
\mathbb{K}^{\bullet \circ} \arrow[ur, no head, "\approx\ \vee"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathbb{K}_{\bullet} \arrow[rr, bend left=10, "\mathrm{DP}^{*}"] \arrow[dd, leftrightarrow, "\vee\ \circ"'] & & \mathbb{K}_{*} \arrow[ll, bend left=10, "\mathrm{ND}_{\bullet}"'] \arrow[dd, leftrightarrow, "\vee\ \circ"] \\
& \mathbb{K} \arrow[ul, "\alpha_{\bullet}"] \arrow[ur, "\alpha_{*}"'] \arrow[dl, "\alpha^{\bullet}\ \circ"] \arrow[dr, "\alpha^{*}\ \circ"'] & \\
\mathbb{K}^{\bullet} \arrow[rr, bend right=10, "\mathrm{DP}^{*}_{\bullet}"'] & & \mathbb{K}^{*} \arrow[ll, bend right=10, "\mathrm{ND}^{\bullet}"]
\end{tikzcd}

\begin{tikzcd}
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathbb{K}, \mathcal{B}) \arrow[d, "\wr\ \mathrm{restr}"'] \arrow[ddr, bend left=20, "F \mapsto F(\mathbb{C}^{*})"] & \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{A}, \mathcal{B}) \arrow[d, "\wr\ F \mapsto F(\mathbb{C}^{\bullet})"'] & \\
\mathcal{B}^{\bullet} \arrow[r, hook, "\mathrm{DP}"] & \mathcal{B}^{*}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& \mathbb{C}^0 \arrow[dl, "d"'] \arrow[dr, "u"] & & \\
\mathbb{C}^1 \arrow[rr, "u'"'] & & L \arrow[dr, "v"] & \\
& & & \mathbb{C}^0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=normal, nodes={font=\small}]
(F(X) \otimes F(Y)) \otimes F(Z) \arrow[r, "{u_{X,Y} \otimes F(Z)}"] \arrow[d, "\wr\ a^{\otimes}_{F(X),F(Y),F(Z)}"'] & F(X \otimes Y) \otimes F(Z) \arrow[d, "u_{X \otimes Y,Z}"] \\
F(X) \otimes (F(Y) \otimes F(Z)) \arrow[d, "F(X) \otimes u_{Y,Z}"'] & F((X \otimes Y) \otimes Z) \arrow[d, "\wr\ F(a^{\mathcal{B}'}_{X,Y,Z})"] \\
F(X) \otimes F(Y \otimes Z) \arrow[r, "u_{X,Y \otimes Z}"'] & F(X \otimes (Y \otimes Z))
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=normal, nodes={font=\small}]
F(X) \otimes \mathbf{1}_{\mathcal{B}} \arrow[r, "F(X) \otimes \alpha"] \arrow[ddr, "\wr\ \delta^{\mathcal{B}}_{F(X)}"'] & F(X) \otimes F(\mathbf{1}_{\mathcal{B}'}) \arrow[d, "u_{X,\mathbf{1}_{\mathcal{B}'}}"] \\
 & F(X \otimes \mathbf{1}_{\mathcal{B}'}) \arrow[d, "\wr\ F(\delta^{\mathcal{B}'}_{X})"] \\
 & F(X)
\end{tikzcd}

\begin{tikzcd}[column sep=large, row sep=normal, nodes={font=\small}]
F(X) \otimes F(Y) \arrow[r, "u_{X,Y}"] \arrow[d, "\sigma^{\mathcal{B}}_{F(X),F(Y)}"'] & F(X \otimes Y) \arrow[d, "F(\sigma^{\mathcal{B}'}_{X,Y})"] \\
F(Y) \otimes F(X) \arrow[r, "u_{Y,X}"'] & F(Y \otimes X)
\end{tikzcd}

\begin{tikzcd}[column sep=large, row sep=normal, nodes={font=\small}]
(C^{i} \otimes_{\mathcal{B}} C^{j}) \otimes C^{*}_{ij} \arrow[d] & (C^{i} \otimes_{\mathcal{B}} C^{j}) \otimes C^{*}_{i+1,j} \arrow[l, "d^{*}_{i+1,j}"'] \arrow[d, "d^{i} \otimes -\,\otimes -"] \\
C^{*} & (C^{i+1} \otimes_{\mathcal{B}} C^{j}) \otimes C^{*}_{i+1,j} \arrow[l, "d"]
\end{tikzcd}

\begin{tikzcd}[column sep=large, row sep=small]
F(X) \otimes_{\mathcal{B}} F(Y) \arrow[r, "?"] \arrow[d] & F(X * Y) \arrow[d] \\
F(I) \otimes_{\mathcal{B}} F(J) \arrow[r] & F(I * J)
\end{tikzcd}

\begin{tikzcd}[column sep=huge, row sep=large]
\Gamma^{pqr}(X) \arrow[r, "\alpha^{X}_{p,qr}"] \arrow[d, "\alpha^{X}_{pq,r}"'] & \Gamma^{p}(\Gamma^{qr}(X)) \arrow[d, "{\Gamma^{p}(\alpha^{\Gamma^{r}(X)}_{q,r})}"] \\
\Gamma^{pq}(\Gamma^{r}(X)) \arrow[r, "\alpha^{\Gamma^{r}(X)}_{p,q}"'] & \Gamma^{p}(\Gamma^{q}(\Gamma^{r}(X)))
\end{tikzcd}

\begin{tikzcd}
 & \Gamma^{q} Y \arrow[r, "g"] & Z \\
\Gamma^{q}(\Gamma^{p}(X)) \arrow[ur, "\Gamma^{q}(f)"] & & \\
\Gamma^{pq}(X) \arrow[u, "\alpha^{X}_{q,p}"'] & &
\end{tikzcd}