\begin{tikzcd}[column sep=normal]
C_0 \arrow[r, bend left=30, "S"] \arrow[r, bend right=30, "S'"'] & C
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
(P_u \otimes P_v) \otimes P_w \arrow[r, "\varphi_{u,v} \otimes \mathrm{id}"]
\arrow[d, "\mathrm{can}"'] & P_{uv} \otimes P_w \arrow[dd, "\varphi_{uv,w}"] \\
P_u \otimes (P_v \otimes P_w) \arrow[d, "\mathrm{id} \otimes \varphi_{v,w}"'] & \\
P_u \otimes P_{vw} \arrow[r, "\varphi_{u,vw}"'] & P_{uvw}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
(P_u \otimes P_v) \otimes P_w \arrow[r, "\varphi_{u,v} \otimes \mathrm{id}"]
\arrow[d, "\mathrm{can}"'] & P_{uv} \otimes P_w \arrow[dd, "\varphi_{uv,w}"] \\
P_u \otimes (P_v \otimes P_w) \arrow[d, "\mathrm{id} \otimes \varphi_{v,w}"'] & \\
P_u \otimes P_{vw} \arrow[r, "\varphi_{u,vw}"'] & P_{uvw}
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=normal]
P_u \otimes P_v \arrow[r, "\varphi_{u,v}"]
\arrow[d, "\lambda(u) \otimes \lambda(v)"'] & P_{uv} \arrow[d, "\lambda(uv)"] \\
P'_u \otimes P'_v \arrow[r, "\varphi'_{u,v}"'] & P'_{uv}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
G \arrow[d] & \Omega S \arrow[l] \arrow[d] & \Omega Y \arrow[l] \arrow[d] &
\Omega G \arrow[l] \arrow[d] & \Omega^{2} S \arrow[l] \arrow[d] &
\Omega^{2} Y \arrow[l] \arrow[d] & \Omega^{2} G \arrow[l] \arrow[d] \\
e \arrow[d] & F \arrow[l] \arrow[d] & F \arrow[l, Rightarrow] \arrow[d] &
e \arrow[l] \arrow[d] & \Omega F \arrow[l] \arrow[d] &
\Omega F \arrow[l, Rightarrow] \arrow[d] & e \arrow[l] \arrow[d] \\
X \arrow[d] & Z \arrow[l] \arrow[d] & G \arrow[l] \arrow[d] &
\Omega X \arrow[l] \arrow[d] & \Omega Z \arrow[l] \arrow[d] &
\Omega G \arrow[l] \arrow[d] & \Omega^{2} X \arrow[l] \arrow[d] \\
S & Y \arrow[l] & G \arrow[l] & \Omega S \arrow[l] & \Omega Y \arrow[l] &
\Omega G \arrow[l] & \Omega^{2} S \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
e \arrow[d] & X_{12} \arrow[l] \arrow[d] & \\
e \arrow[d] & X_{10} \arrow[l] \arrow[d] & \bullet \arrow[l] \arrow[d] \\
X_{00} & X_{10} \arrow[l] & X_{11} \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
S \arrow[d] & X' \arrow[l] \arrow[d] & F' \arrow[l] \arrow[d] &
\Omega S \arrow[l] \arrow[d] & \Omega X' \arrow[l] \arrow[d] &
\Omega F' \arrow[l] \arrow[d] & \Omega^{2} S \arrow[l] \arrow[d] \\
F \arrow[d] & G \arrow[l] \arrow[d] & \Phi \arrow[l] \arrow[d] &
\Omega F \arrow[l] \arrow[d] & \Omega G \arrow[l] \arrow[d] &
\Omega\Phi \arrow[l] \arrow[d] & \phantom{e} \\
X \arrow[d] & Z \arrow[l] \arrow[d] & G' \arrow[l] \arrow[d] &
\Omega X \arrow[l] \arrow[d] & \Omega Z \arrow[l] \arrow[d] &
\Omega G' \arrow[l] & \phantom{e} \\
S & X' \arrow[l] & F' \arrow[l] & \Omega S \arrow[l] &
\Omega X' \arrow[l] & \Omega F' \arrow[l] & \phantom{e}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X_{20} \arrow[d] & & & & & & & & & \\
X_{10} \arrow[d] & & & & & & & & & \\
X_{00} & X_{01} \arrow[l] & X_{02} \arrow[l] &
\Omega X_{00} \arrow[l] \arrow[d] & \Omega X_{01} \arrow[l] \arrow[d] &
\Omega X_{02} \arrow[l] \arrow[d] & \Omega^{2} X_{00} \arrow[l] \arrow[d] &
& & \\
& & & X_{20} \arrow[d] & X_{21} \arrow[l] \arrow[d] &
X_{22} \arrow[l] \arrow[d] & \Omega X_{20} \arrow[l] \arrow[d] & & & \\
& & & X_{10} \arrow[d] & X_{11} \arrow[l] \arrow[d] &
X_{12} \arrow[l] \arrow[d] & \Omega X_{10} \arrow[l] \arrow[d] & & & \\
& & & X_{00} & X_{01} \arrow[l] & X_{02} \arrow[l] &
\Omega X_{00} \arrow[l] \arrow[d] & & & \\
& & & & & & X_{20} \arrow[d] & & & \\
& & & & & & X_{10} \arrow[d] & & & \\
& & & & & & X_{00} & X_{01} \arrow[l] & X_{02} \arrow[l] &
\Omega X_{00} \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& (X \otimes Y) \otimes (Z \otimes T) \arrow[dr, "a_{X \otimes Y{,}Z{,}T}"]
& \\
X \otimes (Y \otimes (Z \otimes T)) \arrow[ur, "a_{X{,}Y{,}Z \otimes T}"]
\arrow[d, "\mathrm{id}_{X} \otimes a_{Y{,}Z{,}T}"'] & &
((X \otimes Y) \otimes Z) \otimes T \\
X \otimes ((Y \otimes Z) \otimes T) \arrow[rr, "a_{X{,}Y \otimes Z{,}T}"']
& & (X \otimes (Y \otimes Z)) \otimes T
\arrow[u, "a_{X{,}Y{,}Z} \otimes \mathrm{id}_{T}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& (X \otimes Y) \otimes Z \arrow[rr, "c_{X \otimes Y{,}Z}"] & &
Z \otimes (X \otimes Y) \arrow[dr, "a_{Z{,}X{,}Y}"] & \\
X \otimes (Y \otimes Z) \arrow[ur, "a_{X{,}Y{,}Z}"]
\arrow[dr, "\mathrm{id}_{X} \otimes c_{Y{,}Z}"'] & & & &
(Z \otimes X) \otimes Y \\
& X \otimes (Z \otimes Y) \arrow[rr, "a_{X{,}Z{,}Y}"'] & &
(X \otimes Z) \otimes Y \arrow[ur, "c_{X{,}Z} \otimes \mathrm{id}_{Y}"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X \otimes (1 \otimes Y) \arrow[rr, "a_{X{,}1{,}Y}"] & &
(X \otimes 1) \otimes Y \\
& X \otimes Y \arrow[ul, "\mathrm{id}_{X} \otimes g_{Y}"]
\arrow[ur, "d_{X} \otimes \mathrm{id}_{Y}"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
FX \otimes (FY \otimes FZ) \arrow[r, "\mathrm{id} \otimes \check{F}"]
\arrow[d, "a'"'] & FX \otimes F(Y \otimes Z) \arrow[r, "\check{F}"] &
F(X \otimes (Y \otimes Z)) \arrow[d, "Fa"] \\
(FX \otimes FY) \otimes FZ \arrow[r, "\check{F} \otimes \mathrm{id}"'] &
F(X \otimes Y) \otimes FZ \arrow[r, "\check{F}"'] &
F((X \otimes Y) \otimes Z)
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=small, nodes={font=\scriptsize}]
FX \otimes FY \arrow[r, "\check{F}"] \arrow[d, "c'"'] &
F(X \otimes Y) \arrow[d, "Fc"] \\
FY \otimes FX \arrow[r, "\check{F}"'] & F(Y \otimes X)
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=small, nodes={font=\scriptsize}]
1' \otimes FX \arrow[r, "\hat{F} \otimes \mathrm{id}_{FX}"] &
F1 \otimes FX \arrow[d, "\check{F}"] \\
FX \arrow[u, "g'_{FX}"] \arrow[r, "Fg_{X}"'] & F(1 \otimes X)
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=small, nodes={font=\scriptsize}]
FX \otimes 1' \arrow[r, "\mathrm{id}_{FX} \otimes \hat{F}"] &
FX \otimes F1 \arrow[d, "\check{F}"] \\
FX \arrow[u, "d'_{FX}"] \arrow[r, "Fd_{X}"'] & F(X \otimes 1)
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=small, nodes={font=\scriptsize}]
FX \otimes FY \arrow[r, "\check{F}"]
\arrow[d, "\lambda_{X} \otimes \lambda_{Y}"'] &
F(X \otimes Y) \arrow[d, "\lambda_{X \otimes Y}"] \\
GX \otimes GY \arrow[r, "\check{G}"'] & G(X \otimes Y)
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\Pi_{0}(P) \arrow[rr] & & \Pi_{0}(P') \\
& M \arrow[ul, "\varepsilon_{0}"] \arrow[ur, "\varepsilon'_{0}"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\Pi_{1}(P) \arrow[rr] & & \Pi_{1}(P') \\
& N \arrow[ul, "\varepsilon_{1}"] \arrow[ur, "\varepsilon'_{1}"'] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \otimes T(A'_{1} \otimes C'_{1})
\arrow[r, "\mathrm{id} \otimes \check{T}^{-1}"]
\arrow[d, "\mathrm{id} \otimes Tu'"'] &
A \otimes (TA'_{1} \otimes TC'_{1}) \arrow[r, "a"] &
(A \otimes TA'_{1}) \otimes TC'_{1} \arrow[r, "u_{1} \otimes \mathrm{id}"]
& (B \otimes TB'_{1}) \otimes TC'_{1} \arrow[d, "a^{-1}"] \\
A \otimes T(A'_{2} \otimes C'_{2})
\arrow[d, "\mathrm{id} \otimes \check{T}^{-1}"'] & & &
B \otimes (TB'_{1} \otimes TC'_{1}) \arrow[d, "\mathrm{id} \otimes \check{T}"] \\
A \otimes (TA'_{2} \otimes TC'_{2}) \arrow[d, "a"'] & & &
B \otimes T(B'_{1} \otimes C'_{1}) \arrow[d, "\mathrm{id} \otimes Tv'"] \\
(A \otimes TA'_{2}) \otimes TC'_{2}
\arrow[r, "u_{2} \otimes \mathrm{id}"'] &
(B \otimes TB'_{2}) \otimes TC'_{2} \arrow[r, "a^{-1}"'] &
B \otimes (TB'_{2} \otimes TC'_{2})
\arrow[r, "\mathrm{id} \otimes \check{T}^{-1}"'] &
B \otimes T(B'_{2} \otimes C'_{2})
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \otimes T(A' \otimes B'')
\arrow[r, "\mathrm{id} \otimes \check{T}^{-1}"]
\arrow[dddd, "\omega"'] &
A \otimes (TA' \otimes TB'') \arrow[r, "a"] &
(A \otimes TA') \otimes TB'' \arrow[r, "u \otimes \mathrm{id}"] &
(B \otimes TB') \otimes TB'' \arrow[d, "a^{-1}"] \\
& & & B \otimes (TB' \otimes TB'') \arrow[d, "\mathrm{id} \otimes c"] \\
& & & B \otimes (TB'' \otimes TB') \arrow[d, "a"] \\
& & & (B \otimes TB'') \otimes TB' \arrow[d, "v \otimes \mathrm{id}"] \\
C \otimes T(B' \otimes C'') &
C \otimes (TB' \otimes TC'') \arrow[l, "\mathrm{id} \otimes \check{T}"] &
C \otimes (TC'' \otimes TB') \arrow[l, "\mathrm{id} \otimes c"] &
(C \otimes TC'') \otimes TB' \arrow[l, "a^{-1}"]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
(A \otimes TA') \otimes (E \otimes TE') \arrow[r, "u \otimes v"]
\arrow[d, "a"'] &
(B \otimes TB') \otimes (F \otimes TF') \arrow[d, "a"] \\
((A \otimes TA') \otimes E) \otimes TE'
\arrow[d, "a^{-1} \otimes \mathrm{id}"'] &
((B \otimes TB') \otimes F) \otimes TF'
\arrow[d, "a^{-1} \otimes \mathrm{id}"] \\
(A \otimes (TA' \otimes E)) \otimes TE'
\arrow[d, "(\mathrm{id} \otimes c) \otimes \mathrm{id}"'] &
(B \otimes (TB' \otimes F)) \otimes TF'
\arrow[d, "(\mathrm{id} \otimes c) \otimes \mathrm{id}"] \\
(A \otimes (E \otimes TA')) \otimes TE'
\arrow[d, "a \otimes \mathrm{id}"'] &
(B \otimes (F \otimes TB')) \otimes TF'
\arrow[d, "a \otimes \mathrm{id}"] \\
((A \otimes E) \otimes TA') \otimes TE' \arrow[d, "a^{-1}"'] &
((B \otimes F) \otimes TB') \otimes TF' \arrow[d, "a^{-1}"] \\
(A \otimes E) \otimes (TA' \otimes TE')
\arrow[d, "\mathrm{id} \otimes \check{T}"'] &
(B \otimes F) \otimes (TB' \otimes TF')
\arrow[d, "\mathrm{id} \otimes \check{T}"] \\
(A \otimes E) \otimes T(A' \otimes E') \arrow[r, "W"'] &
(B \otimes F) \otimes T(B' \otimes F')
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \otimes (TA'_{0} \otimes TA') \arrow[r, "\mathrm{id} \otimes \check{T}"]
\arrow[d, "a"'] & A \otimes T(A'_{0} \otimes A') \arrow[d, "t_{A}"] \\
(A \otimes TA'_{0}) \otimes TA' \arrow[r, "c \otimes \mathrm{id}"'] &
(TA'_{0} \otimes A) \otimes TA'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \otimes (TA'_{0} \otimes TA') \arrow[r, "\mathrm{id} \otimes \check{T}"]
\arrow[d, "a"'] & A \otimes T(A'_{0} \otimes A') \arrow[d, "r_{A}"] \\
(A \otimes TA'_{0}) \otimes TA' \arrow[r, no head] &
(A \otimes TA'_{0}) \otimes TA'
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=small, nodes={font=\scriptsize}]
C \arrow[r, "S"] \arrow[d, "j"'] & C \arrow[d, "j"] \\
Q \arrow[r, "q"'] & Q
\end{tikzcd}