\begin{tikzcd}
\mathcal{V} \arrow[r, "\varphi"] \arrow[dr, "A"'] & \mathcal{V}^{A} \arrow[r, "\psi"] & k\text{-mod gr.} \\
 & C \arrow[ur, "\omega"'] &
\end{tikzcd}

\begin{tikzcd}
 & X \times Y \times Z \arrow[dl, "p_{12}"'] \arrow[dr, "p_{23}"] \arrow[dd, "p_{13}"] & \\
X \times Y & & Y \times Z \\
 & X \times Z &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & X \times Y \times Z \times T \arrow[dl, "\alpha'"'] \arrow[drrr, "\nu'"] & & & & \\
 & X \times Y \times Z \arrow[dl, "\lambda"'] \arrow[d, "\mu"] \arrow[dr, "\nu"] & & & & X \times Z \times T \arrow[dlll, "\alpha"'] \arrow[d, "\gamma"] \arrow[dr, "\beta"] & \\
X \times Y & Y \times Z & X \times Z & & & X \times T & Z \times T
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & X \times Y \times Z \times T \arrow[dl, "n'"'] \arrow[dr, "b'"] & & \\
 & X \times Y \times T \arrow[dl, "a"'] \arrow[d, "c"] \arrow[dr, "b"] & & Y \times Z \times T \arrow[dl, "n"'] \arrow[d, "l"] \arrow[dr, "m"] & \\
X \times Y & X \times T & Y \times T & Y \times Z & Z \times T
\end{tikzcd}

\begin{tikzcd}
 & X \times X \times Y \arrow[dl, "p_{21}"'] \arrow[dr, "p_{32}"] \arrow[dd, no head] & \\
X \times X & & X \times Y \\
 & X \times Y &
\end{tikzcd}

\begin{tikzcd}
X \times Y \arrow[r, "\delta_{X} \times \mathrm{id}_{Y} = q"] \arrow[d, "p"'] & X \times X \times Y \arrow[d, "p_{21}"] \\
X \arrow[r, "\delta_{X}"] & X \times X
\end{tikzcd}

\begin{tikzcd}
 & X \times Y \times P \arrow[dl] \arrow[dr] & \\
X \times P \arrow[dr, no head] & & Y \times P \arrow[dl, no head] \\
 & P &
\end{tikzcd}

\begin{tikzcd}
 & X \times Y \arrow[dl] \arrow[dr, hook] & \\
X \arrow[dr, hook, "\mathrm{diag.}"'] & & X \times X \times Y \arrow[dl, "\mathrm{proj}"] \\
 & X \times X &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & X \arrow[dl, hook, "\Gamma_{gf}"'] \arrow[dr, hook, "\Gamma_{f}"] & & \\
 & X \times Z \arrow[dl, "\mu' = p_{X}"'] \arrow[dr, hook, "\lambda' = \Gamma_{f} \times Z"] & & X \times Y \arrow[dl, hook, "\lambda = \mathrm{id}_{X} \times \Gamma_{g}"'] \arrow[dr, "\mu = p_{Y}"] & \\
X \arrow[dr, hook, "\Gamma_{f}"'] & & X \times Y \times Z \arrow[dl, "p_{21}"'] \arrow[dr, "p_{32}"] \arrow[dd, "p_{31}"] & & Y \arrow[dl, hook, "\Gamma_{g}"] \\
 & X \times Y & & Y \times Z & \\
 & & X \times Z & &
\end{tikzcd}

\begin{tikzcd}
 & X \arrow[d, "\delta"'] & \\
 & X \times X \arrow[dl, "p_{1}"'] \arrow[dr, "p_{2}"] & \\
X & & X
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & & X \times Y \times Z \arrow[dl, "\lambda = p_{XY}"'] \arrow[dr, "\mu = p_{YZ}"] & & \\
 & X \times Y \arrow[dl, "p_{1}"'] \arrow[dr, "p_{2}"] & & Y \times Z \arrow[dl, "q_{1}"'] \arrow[dr, "q_{2}"] & \\
X & & Y & & Z
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & X \times Y \times Z \arrow[dl, "p_{XY}"'] \arrow[d, "p_{YZ}"] \arrow[dr, "p_{XZ}"] & & \\
X \times Y & Y \times Z & X \times Z \arrow[dl, "r_{1}"'] \arrow[dr, "r_{2}"] & \\
 & X & & Z
\end{tikzcd}

\begin{tikzcd}
 & X \arrow[d, "\Gamma_{f}"] & \\
 & X \times Y \arrow[dl, "p_{1}"'] \arrow[dr, "p_{2}"] & \\
X & & Y
\end{tikzcd}

\begin{tikzcd}
A(X) \arrow[r, bend left=20, "u_{*}"] \arrow[d, "k_{X}"'] & A(Y) \arrow[l, bend left=20, "u^{*}"] \arrow[d, "k_{Y}"] \\
k & k
\end{tikzcd}

\begin{tikzcd}
 & X \times Y \arrow[dl, "p_{1}"'] \arrow[dr, "p_{2}"] & \\
X \arrow[dr, no head, "f"'] & & Y \arrow[dl, no head, "g"] \\
 & e &
\end{tikzcd}

\begin{tikzcd}
A \times B \arrow[r, "s_{A,B}"] \arrow[d, "f \times g"'] & B \times A \arrow[d, "g \times f"] \\
A' \times B' \arrow[r, "s_{A',B'}"] & B' \times A'
\end{tikzcd}

\begin{tikzcd}
X \arrow[d, "f"'] & X \times S \arrow[l, "q"'] \arrow[d, "f'"] \\
Y & Y \times S \arrow[l, "p"]
\end{tikzcd}

\begin{tikzcd}
B \arrow[r, "q^{*}(b) = b \otimes 1"] & B \otimes R \\
A \arrow[u, "f^{*}"] \arrow[r, "p^{*}(a) = a \otimes 1"'] & A \otimes R \arrow[u, "f^{*} \otimes \mathrm{id}_{R} = f'^{*}"']
\end{tikzcd}

\begin{tikzcd}
 & X \arrow[dl, "\Gamma_{gf}"'] \arrow[dr, "\Gamma_{f}"] & \\
X \times Z \arrow[dr, "\Gamma_{f} \times \mathrm{id}_{Z} = \lambda"'] & & X \times Y \arrow[dl, "\mathrm{id}_{X} \times \Gamma_{g} = \lambda'"] \\
 & X \times Y \times Z &
\end{tikzcd}

\begin{tikzcd}
 & A & \\
A \otimes C \arrow[ur] & & A \otimes B \arrow[ul] \\
 & A \otimes B \otimes C \arrow[ul, "\lambda^{*}"] \arrow[ur, "\lambda'^{*}"'] &
\end{tikzcd}

\begin{tikzcd}
  & X^{I \sqcup I'} \arrow[dl, "\alpha"'] \arrow[dr, "\alpha'"] & \\
  X^{I} & & X^{I'}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & & & X^{I \sqcup I' \sqcup I''} \arrow[dll, "p_{21}"'] \arrow[d, "p_{31}"] \arrow[drr, "p_{32}"] & & & \\
  & X^{I \sqcup I'} \arrow[dl, no head, "\gamma"'] \arrow[dr, "\gamma'"] & & X^{I \sqcup I''} \arrow[ddl, no head, "\beta"'] \arrow[ddr, no head, "\beta''"] & & X^{I' \sqcup I''} \arrow[dl, no head, "\alpha'"'] \arrow[dr, "\alpha''"] & \\
  X^{I} & & X^{I'} & & X^{I'} & & X^{I''} \\
  & & X^{I} & & X^{I''} & &
\end{tikzcd}

\begin{tikzcd}
  & X^{I \sqcup I' \sqcup I''} \arrow[dl, "p_{21}"'] \arrow[dr, "p_{32}"] & & \\
  X^{I \sqcup I'} \arrow[dr, "\gamma'"'] & & X^{I' \sqcup I''} \arrow[dl, "\alpha'"] \arrow[dr, no head, "\alpha''"] & \\
  & X^{I'} & & X^{I''}
\end{tikzcd}

\begin{tikzcd}
  & X^{I} \arrow[d, "\delta"] & \\
  & X^{I \sqcup I} \arrow[dl, "p_{1}"'] \arrow[dr, "p_{2}"] & \\
  X^{I} & & X^{I}
\end{tikzcd}

\begin{tikzcd}
  X^{s} \arrow[r, hook] & X^{k} \\
  & X^{s'} \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}
  V^{s} \arrow[r] & V^{k} \\
  V \arrow[u] \arrow[r] & V^{s'} \arrow[u]
\end{tikzcd}

\begin{tikzcd}
  & U \arrow[dr, no head, "p"] & \\
  U \cap V \arrow[ur, no head] \arrow[dr, no head] & & W \\
  & V \arrow[ur, no head, "q"'] &
\end{tikzcd}

\begin{tikzcd}
  & U' \arrow[dr, no head, "p-r"] & & \\
  U' \cap V' \arrow[ur, no head, "q-r"] \arrow[dr, no head, "p-r"'] & & U' + V' \arrow[r, no head, "r"] & W' \\
  & V' \arrow[ur, no head, "q-r"'] & &
\end{tikzcd}