\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\widetilde{x} \arrow[r, "\alpha_x"] & x
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\widetilde{x} \arrow[d, "\alpha_x"'] \arrow[dr, "f\alpha_x"] & \widetilde{y}
\arrow[d, "\alpha_y"] \\
x \arrow[r, "f"'] & y
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\widetilde{x} \arrow[r, "\widetilde{f}"] \arrow[d, no head] &
\widetilde{y}\,' \arrow[d, "\alpha'_y"] \\
\widetilde{x} \arrow[r, "f\alpha_x"'] & y
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x' \arrow[r, bend left=20, "\varphi"] \arrow[r, bend right=20, "\psi"'] & y'
\arrow[r, "\underset{\sim}{\alpha}"] & y
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
W_{\mathrm{cof}}^{-1}\mathcal{C}_{\mathrm{cof}} \arrow[r, bend left=20, "\Psi"]
& W^{-1}\mathcal{C} \arrow[l, bend left=20, "\Phi"]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\widetilde{x} \arrow[r, "\widetilde{f}"] \arrow[d, "\underset{\sim}{\alpha_x}"'] &
\widetilde{y} \arrow[r, "i_f"] \arrow[d, "\underset{\sim}{\alpha_y}"] &
\widetilde{y}\,' \arrow[dl, "\beta_f"] \\
x \arrow[r, "f"'] & y &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x \arrow[r, bend left=20, "\varphi"] \arrow[r, bend right=20, "\psi"'] & y
\arrow[r, "\underset{\sim}{f}"] & y'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
a \arrow[r, "f"] \arrow[d, "g"'] & b \\
c &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x \arrow[r, "f"] & \widetilde{y} & y \arrow[l, "\underset{\sim}{i}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& \widetilde{y} \arrow[r, "g"] & \widetilde{z} \arrow[d, "\underset{\sim}{k}"] & z
\arrow[l, "\underset{\sim}{j}"'] \\
& & \widetilde{z}\,' &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x \arrow[r, "1"] & \widetilde{x} & x \arrow[l, "\underset{\sim}{1}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\emptyset \arrow[r] \arrow[d] & x \arrow[r, "\underset{\sim}{i}"] \arrow[d, "\underset{\sim}{f}"'] & \bar{x} \arrow[dl, "\alpha", dashed] \\
y \arrow[r, "1"'] & y \arrow[ur, "g"', dashed] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x \arrow[r, "\underset{\sim}{i}"] \arrow[d, "1_x"'] \arrow[dr, "gf"] & \bar{x}
\arrow[d, "\underset{\sim}{\alpha}"] \\
x \arrow[r, "f"'] & y
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\widetilde{\mathcal{C}} \arrow[r] & W_0^{-1}\mathcal{C} \arrow[r] &
W^{-1}\mathcal{C}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x \arrow[r, "f"] & \widetilde{y} & y \arrow[l, "\underset{\sim}{i}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x \arrow[r, "f_0"] & \bar{y} \arrow[r, "\underset{\sim}{j}"] & \widetilde{y}\,'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x \arrow[r] \arrow[d, "i_1"'] & y' \arrow[d] \\
x \vee x \arrow[r] \arrow[d] & y' \vee x \arrow[d] \\
x \vee y'' \arrow[r] & y' \vee y''
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x \vee x \arrow[r, "(f_0) \vee (f'_0)"] \arrow[d, "\nabla_x"'] & \bar{y}
\amalg \bar{y}\,' \arrow[d, dashed] \arrow[dr, "{(\alpha, \alpha')}"] & \\
x \arrow[r, "g"', dashed] & \widetilde{y} \arrow[r, "\varphi_0"'] & \bar{y}\,''
\arrow[d, "\alpha''"] \\
& & y
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
x \arrow[rr, "f"] \arrow[dr, "f_0"'] \arrow[ddr, "f'_0"'] & & y \\
& \bar{y} \arrow[ur, "\alpha'"'] \arrow[d, "u"'] & \\
& \bar{y}\,'' \arrow[uur, "\alpha''"'] &
\end{tikzcd}