\begin{tikzcd}
B \otimes B \otimes C & B \otimes C \arrow[l, "\mathrm{id}_B \otimes q_C"'] \\
B \otimes C \arrow[u, "p_B \otimes \mathrm{id}_C"] & C \arrow[l, "q_C"] \arrow[u, "q_C"']
\end{tikzcd}

\begin{tikzcd}
A \otimes C = C & B \otimes C \arrow[l, "\varepsilon \otimes \mathrm{id}_C"'] \\
 & C \arrow[ul, "\sim"] \arrow[u, "q_C"']
\end{tikzcd}

\begin{tikzcd}
C \otimes C & B \otimes C \arrow[l, "{(g_C, p_2)}"'] & C \arrow[l, "q_C"'] \arrow[ll, bend left=30, "p_1"]
\end{tikzcd}

\begin{tikzcd}
B \otimes C & C \otimes C \arrow[l, "{(q_C, p_2)}"'] & B \arrow[l, "g_C"'] \arrow[ll, bend left=30, "p_1"]
\end{tikzcd}

\begin{tikzcd}
B \otimes C & C \arrow[l, "q_C"'] \\
C \arrow[u, "\mathrm{pr}_2"] & \mathcal{D} \arrow[l, "h"] \arrow[u, "h"']
\end{tikzcd}

\begin{tikzcd}
\mathcal{U} \arrow[r, "\Delta_{\mathcal{U}}"] \arrow[rrr, bend right=20, "v * w"'] & \mathcal{U} \otimes \mathcal{U} \arrow[r, "v \otimes w"] & C \otimes C \arrow[r, "\Delta_C"] & C
\end{tikzcd}

\begin{tikzcd}
E \times E \arrow[r, "{(g, p_2)}"] \arrow[rr, bend right=30, "p_1"'] & G \times E \arrow[r, "q_E"] & E
\end{tikzcd}

\begin{tikzcd}
G \times E \arrow[r, "{(q_E, p_2)}"] \arrow[rr, bend right=30, "p_1"'] & E \times E \arrow[r, "g_E"] & G
\end{tikzcd}

\begin{tikzcd}
C \arrow[r, "q_C"] \arrow[rr, bend right=30, "p_1"'] & B \otimes C \arrow[r, "{(g_C, p_2)}"] & C \otimes C
\end{tikzcd}

\begin{tikzcd}[column sep=large]
B \arrow[r, "g_C"] \arrow[rr, bend right=30, "p_1"'] & C \otimes C \arrow[r, "{\varphi = (q_C, p_2)}"] & B \otimes C
\end{tikzcd}

\begin{tikzcd}
C \arrow[r, "q_C"] \arrow[rrr, bend right=20, "\psi(v)"'] & B \otimes C \arrow[r, "v \otimes \mathrm{id}_C"] & C \otimes C \arrow[r, "\Delta_C"] & C
\end{tikzcd}

\begin{tikzcd}
B \arrow[r, "g_C"] \arrow[rrr, bend right=20, "\psi'(w)"'] & C \otimes C \arrow[r, "w \otimes \mathrm{id}_C"] & C \otimes C \arrow[r, "\Delta_C"] & C
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
C \arrow[d, "q_C"] & & C \arrow[d, "p_1"] \\
B \otimes C \arrow[d, "g_C \otimes \mathrm{id}_C"] & & C \otimes C \arrow[d, "w \otimes \mathrm{id}_C"] \\
C \otimes C \otimes C \arrow[d, "w \otimes \mathrm{id}_{C \otimes C}"] & & C \otimes C \arrow[d, "\Delta_C"] \\
C \otimes C \otimes C \arrow[d, "\Delta_C \otimes \mathrm{id}_C"] & & C \\
C \otimes C \arrow[d, "\Delta_C"] & & \\
C & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
B \arrow[d, "g_C"] & & B \arrow[d, "p_1"] \\
C \otimes C \arrow[d, "q_C \otimes \mathrm{id}_C"] & & B \otimes C \arrow[d, "v \otimes \mathrm{id}_C"] \\
B \otimes C \otimes C \arrow[d, "v \otimes \mathrm{id}_{C \otimes C}"] & & C \otimes C \arrow[d, "\Delta_C"] \\
C \otimes C \otimes C \arrow[d, "\Delta_C \otimes \mathrm{id}_C"] & & C \\
C \otimes C \arrow[d, "\Delta_C"] & & \\
C & &
\end{tikzcd}

\begin{tikzcd}
G \times E \arrow[r] \arrow[d, "\wr"'] & G \times E' \arrow[d, "\wr"] \\
E \times E \arrow[r, "\sim"] & E' \times E'
\end{tikzcd}