\begin{tikzcd}
X & X^n_S \arrow[r] & Z \\
Y \arrow[u] & Y^n_S \arrow[u] \arrow[r] & \mathrm{Symm}^n_S(Y) \arrow[u] \\
S & &
\end{tikzcd}

\begin{tikzcd}
G & \\
S \arrow[u, no head] & T \arrow[l]
\end{tikzcd}

\begin{tikzcd}
\struck{G} & G' \arrow[l] & G \arrow[l, "r"'] \\
& S \arrow[u, no head] & T \arrow[l, no head] \arrow[u, no head]
\end{tikzcd}

\begin{tikzcd}
G & G^r_S \arrow[l] & G^r_T \arrow[l] \\
S \arrow[u, no head] & & T
\end{tikzcd}

\begin{tikzcd}
F & G \arrow[l, "\alpha(r)"'] \\
S \arrow[u] & T \arrow[l] \arrow[u]
\end{tikzcd}

\begin{tikzcd}
X \arrow[d, no head] & Y \arrow[l, "\alpha\ (r)"'] \arrow[d] & Z \arrow[l, "\beta\ (s)"'] \arrow[d] \arrow[ll, bend right=30, "(rs)"'] \\
S & T \arrow[l] & U \arrow[l] \\
& F & G \\
& & G \otimes_T F
\end{tikzcd}

\begin{tikzcd}
& X_2 & \\
X & \times \arrow[l] & \\
& X_1 \arrow[d] & Y \arrow[l, "r_1"] \arrow[uul, "r_2"'] \arrow[d] \\
& S & T \arrow[l]
\end{tikzcd}

\begin{tikzcd}
X^n_S \arrow[r, "\varphi"] \arrow[d] & Y \arrow[d] & S \arrow[l, "\mathrm{Can}^{\varphi}_F"'] \\
X'^n_{S'} \arrow[r, "\varphi'"'] & Y' & S' \arrow[l, "\mathrm{Can}^{\varphi'}_{F'}"]
\end{tikzcd}

\begin{tikzcd}
Y & Y' \arrow[l] \\
S \arrow[u, "\mathrm{Can}^{\varphi}_F"] & S' \arrow[l] \arrow[u, "\mathrm{Can}^{\varphi'}_{F'}"']
\end{tikzcd}

\begin{tikzcd}
X^{r_1+r_2}_S \arrow[r, "\varphi"] & Y & S \arrow[l, "\mathrm{Can}^{\varphi}_{F_1 \times F_2}"'] \\
X^{r_1}_S \times_S X^{r_2}_S \arrow[u, "\wr\ \mathrm{canon.}"] \arrow[r, "\varphi_1 \times_S \varphi_2"'] & Y_1 \times_S Y_2 \arrow[u, "u"'] & S \arrow[l, "\mathrm{Can}^{\varphi_1}_{F_1} \times \mathrm{Can}^{\varphi_2}_{F_2}"] \arrow[u, no head]
\end{tikzcd}

\begin{tikzcd}
G & X \arrow[d, "r"] & G \otimes_Y F \\
F & Y \arrow[d, "s"] & \\
& Z &
\end{tikzcd}

\begin{tikzcd}
X^{rs}_Z \arrow[r, "\chi"] & \mathfrak{Z} \\
(X^r_Y)^s_Z \arrow[u, "\mathrm{can}"] \arrow[r, "\varphi^s_Z"'] & \mathfrak{X}^s_Z \arrow[u, "u"']
\end{tikzcd}

\begin{tikzcd}
(G_Y)^{rs}_Y \arrow[r, "\varphi_Y"] \arrow[d, no head] & G'_Y \arrow[d] \\
G^{rs}_Z \arrow[r, "\varphi"'] & G'
\end{tikzcd}

\begin{tikzcd}
(G_Y)^{rs} \arrow[r, "\psi^s_Y"] \arrow[dr, "\mathrm{can}\,\varphi_Y"'] & G''^{s}_Z \arrow[d, "\chi"] \\
& G'
\end{tikzcd}

\begin{tikzcd}[column sep=small, nodes={font=\scriptsize}]
Z \arrow[r, "\mathrm{Can}_{Y/Z}"] \arrow[drr] \arrow[ddrr, "\mathrm{Can}_{X/Z}"'] & \Sigma^s(Y/Z) \arrow[r, "\Sigma^s(\mathrm{Can}_{X/Y}/Z)"] & \Sigma^s(\Sigma^r(X/Y)/Z) \arrow[d, "\mathrm{can}"] \\
& & \Sigma^s(\Sigma^r(X/Z)/Z) \arrow[d, "\mathrm{can}"] \\
& & \Sigma^{sr}(X/Z)
\end{tikzcd}

\begin{tikzcd}
\Sigma^r(X/Y) & Y \arrow[l, "\mathrm{Can}_{X/Y}"'] \\
\Sigma^r(X'/Y') \arrow[u, "\mathrm{can}"] & Y' \arrow[l, "\mathrm{Can}_{X'/Y'}"] \arrow[u]
\end{tikzcd}

\begin{tikzcd}
Y \arrow[r, "{(\mathrm{Can}_{X_1/Y}, \mathrm{Can}_{X_2/Y})}"] \arrow[dr, "\mathrm{Can}_{X/Y}"'] & \Sigma^{r_1}(X_1/Y) \times \Sigma^{r_2}(X_2/Y) \arrow[d] \\
& \Sigma^r(X/Y)
\end{tikzcd}

\begin{tikzcd}
  k \arrow[r, no head, "\mathrm{rad}"] \arrow[d, no head] & k' \arrow[d, no head] \\
  k'' \arrow[r, no head, "\mathrm{rad}"'] & k'k''
\end{tikzcd}

\begin{tikzcd}
  F \arrow[r, hook] & X \arrow[d] \\
  & Y
\end{tikzcd}

\begin{tikzcd}
  X \arrow[d] \\
  Y \arrow[d] \\
  B_F
\end{tikzcd}

\begin{tikzcd}
  Y \arrow[d] \\
  B_F \arrow[d] \\
  B_{\struck{X}}
\end{tikzcd}

\begin{tikzcd}
  B_F \arrow[d] \\
  B_{\struck{X}} \arrow[d] \\
  B_Y
\end{tikzcd}