\begin{tikzcd}[column sep=small]
 & A \times \Delta \arrow[dl, "p_{\Delta}"'] \arrow[dr, "p_A"] & \\
A \arrow[dr, "q_A"'] & & \Delta \arrow[dl, "q_{\Delta}"] \\
 & e &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & (A \times \Delta)^{\wedge} \supset s\underline{A} = \underline{\mathrm{Hom}}(\Delta^{\circ}, A) \arrow[dl, "p_{\Delta*}"'] \arrow[dr, "p_{A*}"] & \\
A^{\wedge} \arrow[dr, "q_{A*}"'] & & \Delta^{\wedge} \arrow[dl, "q_{\Delta*}"] \\
 & (\mathrm{Ens}) &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & A \times B \arrow[dl, "p_B"'] \arrow[dr, "p_A"] & \\
A \arrow[dr, "q_A"'] & & B \arrow[dl, "q_B"] \\
 & e &
\end{tikzcd}

\begin{tikzcd}[column sep=tiny, row sep=large, nodes={font=\small}]
\underline{\mathrm{Hom}}(A^{\circ}, B^{\wedge}) \simeq \arrow[d, "{X' \mapsto \Gamma_B \circ X'}"'] & (A \times B)^{\wedge} \arrow[dl, "(p_B)_*"] \arrow[dr, "(p_A)_*"'] & \simeq \underline{\mathrm{Hom}}(B^{\circ}, A^{\wedge}) \arrow[d, "{X'' \mapsto \Gamma_A \circ X''}"] \\
\underline{\mathrm{Hom}}(A^{\circ}, (\mathrm{Ens})) \simeq A^{\wedge} \arrow[dr, "{q_{A*} = \Gamma_A}"'] & & B^{\wedge} = \underline{\mathrm{Hom}}(B^{\circ}, (\mathrm{Ens})) \arrow[dl, "{q_{B*} = \Gamma_B}"] \\
 & (\mathrm{Ens}) &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & \underline{\mathrm{Hom}}(B^{\circ}, A) \arrow[dl] \arrow[dr, "(\Gamma_A)^{\circ}"] & \\
A \arrow[dr, "\Gamma_A"'] & & B^{\wedge} = \underline{\mathrm{Hom}}(B^{\circ}, (\mathrm{Ens})) \arrow[dl, "\Gamma_B"] \\
 & \phantom{(\mathrm{Ens})} &
\end{tikzcd}

\begin{tikzcd}
 & (A \times B)^{\wedge} \arrow[dl, "p_{B*}"'] \arrow[dr, "p_{A*}"] & \\
A^{\wedge} \arrow[ur, bend left=50, "p_B^{*}"] \arrow[dr, "q_{A*} = \Gamma_A"'] & & B^{\wedge} \arrow[ul, bend right=50, "p_A^{*}"'] \arrow[dl, "q_{B*} = \Gamma_B"] \\
 & (\mathrm{Ens}) &
\end{tikzcd}

\begin{tikzcd}[column sep=large]
X'_0 \arrow[r, hook, "f'_0"] \arrow[d, "{u_0 \,=}"'] & X'_1 \arrow[r, hook, "f'_1"] \arrow[d, "u_1"] & X'_2 \arrow[r, hook, "f'_2"] \arrow[d, "u_2"] & \cdots & X'_\infty \arrow[d, "u_\infty"] \\
X_0 \arrow[r, "f_0"'] & X_1 \arrow[r, "f_1"'] & X_2 \arrow[r, "f_2"'] & \cdots & X_\infty
\end{tikzcd}

\begin{tikzcd}
X_0 \arrow[r, hook] \arrow[dr, "g_0"'] & X_1 \arrow[r, hook] \arrow[d, "g_1"] & X_2 \arrow[r, hook] \arrow[dl, "g_2"] & \cdots & X_n \arrow[r, "f_n"] & X_{n+1} \\
 & Y & & & &
\end{tikzcd}

\begin{tikzcd}[column sep=large, row sep=large]
X \arrow[r, "g"] \arrow[d, "i"'] & Y \arrow[d, "p"] \\
X' \arrow[r, "g'"'] \arrow[ur, dashed, "\varphi"] & Y'
\end{tikzcd}

\begin{tikzcd}
A \arrow[r, "\alpha"] \arrow[d, "\text{cof.}"'] & X \arrow[d, "p"] \\
A \times I \arrow[r, "h"'] \arrow[ur] & Y
\end{tikzcd}

\begin{tikzcd}
A & X \arrow[l] \\
A^{I} \arrow[u, "\partial_0"] & Y \arrow[u, "\text{cofib.}"'] \arrow[l, Rightarrow]
\end{tikzcd}

\begin{tikzcd}
A \arrow[r] \arrow[d] & X \arrow[d] \\
B \arrow[r] \arrow[ur] & Y
\end{tikzcd}

\begin{tikzcd}[column sep=6em, row sep=6em]
X \arrow[r, bend left=35, "g_0 = \varphi i"] \arrow[r, bend right=35, "g_1 = g"'] \arrow[d, "i"'] & Y \arrow[d, "p"] \\
X' \arrow[r, bend left=35, "g'_0 = p\varphi"] \arrow[r, bend right=35, "g'_1 = g'"'] \arrow[ur, bend left=15, "\theta_0 = \varphi"] \arrow[ur, dashed, bend right=15, "\theta_1 = \overline{\varphi}"'] & Y'
\end{tikzcd}

\begin{tikzcd}[column sep=small, nodes={font=\small}]
(X \sqcup X) \arrow[d, "\text{cofib?}"'] & \sqcup & (X' \sqcup X') \arrow[d, "\text{cofib?}"'] & \\
X \otimes J & \sqcup \arrow[d] & X' \otimes J \arrow[r] & (Y \otimes J) \\
Y & \sqcup & Y' &
\end{tikzcd}

\begin{tikzcd}
X \otimes J \arrow[r, "k"] \arrow[d, "i \otimes J"'] & Y \arrow[d, "p"] \\
X' \otimes J \arrow[r, "h"'] & Y'
\end{tikzcd}

\begin{tikzcd}
X \otimes J \arrow[r, "k"] \arrow[d, "i \otimes J"'] & Y \arrow[d, "p"] \\
X' \otimes J \arrow[r, "h"'] & Y'
\end{tikzcd}

\begin{tikzcd}[column sep=large]
X \sqcup X \arrow[d] \arrow[rd, bend left=20, "{u = (g_0,\ g_1)}"] & \\
X \otimes J \arrow[r, dashed, "k'?"] \arrow[d, "i \otimes J"'] & Y \arrow[d, "p"] \\
X' \otimes J \arrow[r, "h"'] & Y'
\end{tikzcd}

\begin{tikzcd}[column sep=large]
X \sqcup X \arrow[r, "{u = (g_0,\ g_1)}"] \arrow[d, "j"'] & Y \arrow[d, "p\ \text{fibration}"] \\
X \otimes J \arrow[r, "v"'] \arrow[ur, dashed, "k'?"] & Y'
\end{tikzcd}

\begin{tikzcd}[column sep=6em, row sep=4em]
X \arrow[r, bend left=35, "g_0 = \varphi i"] \arrow[r, bend right=35, "g_1 = \varphi i"'] \arrow[d] & Y \arrow[d] \\
X' \arrow[r, bend left=35, "g'_0 = g'"] \arrow[r, bend right=35, "g'_1 = p\varphi"'] & Y'
\end{tikzcd}

\begin{tikzcd}[column sep=6em, row sep=6em]
X \arrow[r, bend left=35, "g_0 = \varphi i"] \arrow[r, bend right=35, "g_1 = g"'] \arrow[d, "i"'] & Y \arrow[d, "p"] \\
X' \arrow[r, bend left=35, "g'_0 = p\varphi"] \arrow[r, bend right=35, "g'_1 = g'"'] \arrow[ur, "\varphi" description] & Y'
\end{tikzcd}

\begin{tikzcd}[column sep=4em, row sep=3.5em]
A_i \arrow[r, hook] \arrow[d, hook] \arrow[ddr, bend right=70, "\varphi_i"'] & B_i \arrow[d, hook] & \\
A_{i+1} \arrow[r, hook] \arrow[dr, "\varphi_{i+1}"'] & B'_{i+1} \arrow[r, hook] \arrow[d, "h'_{i+1}"] & B_{i+1} \arrow[dl, dashed, "h_{i+1}?"] \\
 & Y &
\end{tikzcd}