\begin{tikzcd}
U \arrow[r, "\simeq"] & N \times Y_U \\
Y_U \arrow[u, hook] \arrow[r, "\simeq"] & a \times Y_U \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}
X' \cap Y = Y' \arrow[r, hook] \arrow[d, "\simeq"] & X' \arrow[d, "\simeq"] \\
Y' \times \lbrace a \rbrace \arrow[r, hook] & Y' \times Z
\end{tikzcd}

\begin{tikzcd}
Y' \arrow[r, hook] \arrow[d, "\simeq"] & X'_{i_0} \arrow[d, "\simeq"] \\
\lbrace a \rbrace \times Y' \arrow[r, hook] & Z_{i_0} \times Y'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & Z \arrow[dl, "p"'] \arrow[dr, dashed, hook, "i"] & & \\
Y \arrow[dr, "i'"'] & & X \arrow[dl, "p'"] & X - Z \arrow[l, hook'] \arrow[dll, bend left=30, "q"] \\
 & \mathfrak{X} & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
F_1 & \partial F_1 = F_1|\partial\Phi_1 \arrow[l] \\
\Phi_1 & \partial\Phi_1 \arrow[l, "\pi_1"'] \arrow[d, "\overline{\pi}_1"] \\
 & \Phi_0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
V' = V^{\circ\prime} \sqcup_{\partial} U'' \arrow[r] & W^{\circ} \arrow[r]
& W \\
 & V = V^{\circ} \sqcup_{U'} U \arrow[ur] & U \arrow[l, hook']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[r, "f"] & Y & & {|X|} \arrow[r, "{|f|}"] & {|Y|} \\
X' \arrow[u] \arrow[r] & Y' \arrow[u, hook] & & {|X'|} \arrow[u]
\arrow[r] & {|Y'|} \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
Y \arrow[r, "i"] \arrow[d, "j"'] & X \\
Z &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[r, hook, "i"] \arrow[d, hook', "j"'] & Y \arrow[d, hook] \\
Z \arrow[r, hook] & S'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
Y \arrow[r] \arrow[d, hook'] & e \arrow[d, "\sigma_n"] \\
X \arrow[r, "f"'] & I_0^n
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d] & Y \arrow[d, "\Psi"] \\
e \arrow[r] & I_0^p
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d, "{(e,\ \varphi)}"'] & Y \arrow[d, "{(\Psi \times \Phi)}"] \\
e \times I_0^n \simeq I_0^n \arrow[r, hook] & I_0^p \times I_0^n \simeq I_0^{p+n}
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d, "\varphi'"'] & Z \arrow[d, "{(\Psi',\ \Phi')}"] \\
e \times I_0^{n'} \simeq I_0^{n'} \arrow[r, hook] & I_0^{p'} \times I_0^{n'} = I_0^{p'+n'}
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d, "{(\varphi',\ \varphi)}"'] & Z \arrow[d, "{(\Psi',\ \Phi',\ \overline{\varphi})}"] \\
e \times I_0^{n'} \times I_0^n \arrow[r, hook] & I_0^{p'} \times I_0^{n'} \times I_0^n
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d, "{(\varphi,\ \varphi')}"'] & Y \arrow[d, "{(\Psi,\ \Phi,\ \overline{\varphi}')}"] \\
e \times I_0^n \times I_0^{n'} \arrow[r, hook] & I_0^p \times I_0^n \times I_0^{n'}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & Z \arrow[dd] & & \\
X \arrow[ur, hook] \arrow[rr, hook] \arrow[dd] & & Y \arrow[dd] & \\
 & I^{N'} \arrow[rr, dashed] & & S \\
I^{\nu} \arrow[ur, hook] \arrow[rr, hook] & & I^{N} \arrow[ur, dashed] &
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook, "i"] \arrow[d, "p"'] & Y \arrow[d, "p'"] \\
Z \arrow[r, hook, "i'"'] & S
\end{tikzcd}

\begin{tikzcd}
Z \sqcup Y \arrow[r] \arrow[dr, "{(f,\ g)}"'] & S \arrow[d, dashed] \\
 & T
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d] \arrow[dr] & Y \arrow[d] \\
Z \arrow[r, hook] & I^m
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d] & Y \arrow[d, "{\varphi = (\varphi_1, \ldots, \varphi_n)}"] \\
e \arrow[r] & I^n
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d] & Y \arrow[d] \\
X' = I^p \arrow[r, hook] \arrow[d] & Y' = I^{n+p} \arrow[d] \\
e \arrow[r, hook] & S
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook, "i"] \arrow[d, "p"'] & Y \arrow[d, "?"] \\
Z \arrow[r, hook, "?"] & I^n
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d] & Y \arrow[d] \\
e \arrow[r] & I^q
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook, "i"] \arrow[d, "p"'] & Y \arrow[d] \\
Z \arrow[r] \arrow[d] & I^{p+q} = S \arrow[d] \\
e \arrow[r] & I^q = S'
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & Z \arrow[dl, hook'] \arrow[dr, hook] & \\
X & & Y
\end{tikzcd}