\begin{tikzcd}
A \arrow[r, bend left=20, "f"] & B \arrow[l, bend left=20, "g"]
\end{tikzcd}

\begin{tikzcd}
\varphi \arrow[r, "\lambda"] \arrow[rr, bend right=35, "\mathrm{id}_{\varphi}"'] & \varphi^{2} \arrow[r, bend left=18, "\varphi*\alpha"] \arrow[r, bend right=18, "\alpha*\varphi"'] & \varphi
\end{tikzcd}

\begin{tikzcd}
\varphi \arrow[r, "\lambda"] & \varphi^{2} \arrow[r, bend left=18, "\varphi*\lambda"] \arrow[r, bend right=18, "\lambda*\varphi"'] & \varphi^{3}
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, "u"] & \varphi(X) \arrow[r, bend left=18, "\varphi(u)"] \arrow[r, bend right=18, "\lambda(X)"'] & \varphi^{2}(X)
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, "u"] & \varphi(X) \arrow[r, "\alpha(X)"] & X
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, "u"] \arrow[d, "u"'] & \varphi(X) \arrow[d, dashed, "\varphi(u)"] \\
\varphi(X) \arrow[r, "\lambda(X)"'] & \varphi^{2}(X)
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\varphi_{ki} \arrow[r, "\lambda_{kji}"] \arrow[d, no head] & \varphi_{kj}\varphi_{ji} \arrow[d, no head] \\
f_{k} g_{i} & f_{k}\, g_{j} f_{j}\, g_{i}
\end{tikzcd}

\begin{tikzcd}
X' \arrow[r, "u'"] \arrow[rr, bend right=30, "\lambda'(X')"'] & \varphi'(X'') \arrow[r, "\varphi'(u'')"] & \varphi'\varphi''(X')
\end{tikzcd}

\begin{tikzcd}
X'' \arrow[r, "u''"] \arrow[rr, bend right=30, "\lambda''(X'')"'] & \varphi''(X') \arrow[r, "\varphi''(u')"] & \varphi''\varphi'(X'')
\end{tikzcd}

\begin{tikzcd}
C \arrow[r, bend left=20, "F"] & C' \arrow[l, bend left=20, "G"]
\end{tikzcd}

\begin{tikzcd}
S(A) \arrow[r, "S(\varphi(A))"] \arrow[rr, bend right=30, "\text{identité}"'] & S(TS(A)) = (ST)(S(A)) \arrow[r, "\psi(S(A))"] & S(A)
\end{tikzcd}

\begin{tikzcd}
T(X) \arrow[r, "\varphi(T(X))"] \arrow[rr, bend right=30, "\text{identité}"'] & T(ST(X)) = (TS)(T(X)) \arrow[r, "T(\psi(X))"] & T(X)
\end{tikzcd}

\begin{tikzcd}[column sep=small, nodes={font=\scriptsize}]
\mathrm{Hom}_{C}(A,B) \arrow[r] \arrow[rr, bend right=30] & \mathrm{Hom}_{C'}\bigl( S(A), S(B) \bigr) & \mathrm{Hom}_{C}\bigl( A, TS(B) \bigr) \arrow[l, "\simeq"']
\end{tikzcd}

\begin{tikzcd}[column sep=large]
\mathrm{Hom}(\widetilde{A}, \widetilde{X}) \arrow[r, "u \mapsto \widetilde{\eta}\widetilde{u}"] & \mathrm{Hom}(\widetilde{A}, \widetilde{Y}) \\
\mathrm{Hom}(A, T(X)) \arrow[u, "\approx"] \arrow[r, "u \mapsto T(\eta)u"] \arrow[dr, "\rho_A"'] & \mathrm{Hom}(A, T(Y)) \arrow[u, "\approx"'] \\
 & \mathrm{Hom}(A, Y) \arrow[u]
\end{tikzcd}

\begin{tikzcd}
A \arrow[r, "i"] \arrow[dr, "\varepsilon"'] & T(A) \arrow[d, "t"] \\
 & X^{0}
\end{tikzcd}

\begin{tikzcd}
X^{i} \arrow[r, "i_{X^{i}}"] \arrow[dr, "d^{i}"'] & T(X^{i}) \arrow[d, "t"] \\
 & X^{i+1}
\end{tikzcd}

\begin{tikzcd}
X^{n} \arrow[r, "d^{n}"] \arrow[d, "\bar{u}^{n}"'] & X^{n+1} \arrow[d, "\bar{u}^{n+1}"] \\
Y^{n} \arrow[r, "d^{n}"] & Y^{n+1}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & 1 \arrow[d, no head] & \\
 & 2 \arrow[dl, no head] \arrow[d, no head] \arrow[dr, no head] & \\
5 & 3 \arrow[d, no head] & 6 \arrow[d, no head] \\
 & 4 & 7
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
T^{i-1}(B) \arrow[r] \arrow[d] & T^{i-1}(C) \arrow[r] \arrow[d] & T^{i}(A) \arrow[r] \arrow[d] & T^{i}(B) \arrow[r] \arrow[d] & T^{i}(C) \arrow[d] \\
T'^{i-1}(B) \arrow[r] & T'^{i-1}(C) \arrow[r] & T'^{i}(A) \arrow[r] & T'^{i}(B) \arrow[r] & T'^{i}(C)
\end{tikzcd}

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

\begin{tikzcd}
A \arrow[r, bend left=20, "T"] & B \arrow[l, bend left=20, "S"]
\end{tikzcd}

\begin{tikzcd}
A \arrow[r, bend left=20, "T"] & B \arrow[l, bend left=20, "S"]
\end{tikzcd}

\begin{tikzcd}
C' \arrow[r, bend left=20, "T"] & C'/C'' \arrow[l, bend left=20, "S"]
\end{tikzcd}

\begin{tikzcd}
x \arrow[r, hook] & y \\
x'' \arrow[r, hook] \arrow[u, hook] & y'' \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}
\Gamma'x \arrow[r] \arrow[d] & \Lambda\Gamma' \\
R^{0}(\Gamma'/\Gamma'') \arrow[ur] &
\end{tikzcd}

\begin{tikzcd}
A \arrow[r, bend left=20, "T"] & B \arrow[l, bend left=20, dashed, "S"]
\end{tikzcd}

\begin{tikzcd}
F \arrow[r] \arrow[d] & i_{*} i^{*} F \arrow[d] \\
j_{*} j^{*} F \arrow[r] & j_{*} j^{*} i_{*} i^{*} F
\end{tikzcd}

\begin{tikzcd}
F \arrow[r] \arrow[d] & i_{*}(G) \arrow[d] \\
j_{*}(H) \arrow[r] & j_{*}\bigl( \varphi(G) \bigr)
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
C'' \arrow[r, "j_{*}" description] &
C \arrow[l, bend right=45, "j^{*}"'] \arrow[l, bend left=45, "j^{!}"]
  \arrow[r, "i^{*}" description] &
C' \arrow[l, bend right=45, "i_{!}"'] \arrow[l, bend left=45, "i_{*}"]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
0 \arrow[r] & i_{!} i^{*} F \arrow[r] \arrow[d] & F \arrow[r] \arrow[d] &
  j_{*} j^{*} F \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & i_{!} i^{*} F' \arrow[r] \arrow[d] & F' \arrow[r] \arrow[d] &
  j_{*} j^{*} F' \arrow[r] & 0 \\
0 \arrow[r] & i_{!} i^{*} F'' \arrow[r] & F'' & &
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
C'' \arrow[r, "j_{*}" description] &
C \arrow[l, bend right=35, "j^{*}"'] \arrow[r, "i^{*}" description] &
C' \arrow[l, bend right=35, "i_{!}"']
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
C'^{\circ} \arrow[r, "i_{!}^{\circ}" description] &
C^{\circ} \arrow[l, bend right=35, "i^{*\circ}"'] \arrow[r, "j^{*\circ}" description] &
C''^{\circ} \arrow[l, bend right=35, "j_{*}^{\circ}"']
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
C'' \arrow[r, bend left=25, "j_{*}"] &
C \arrow[l, bend left=25, "j^{*}"] \arrow[r, "i^{*}"] &
C'
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
C'' \arrow[r, "j_{*} = j_{!}" description] &
C \arrow[l, bend right=30, "j^{*}"] \arrow[l, bend left=30, "j^{!}"']
  \arrow[r, "i^{*} = i^{!}" description] &
C' \arrow[l, bend right=30, "i_{!}"'] \arrow[l, bend left=30, "i_{*}"]
\end{tikzcd}

\begin{tikzcd}[column sep=huge, nodes={font=\scriptsize}]
C'' \arrow[r, bend left=25, "j^{*}"] &
C \arrow[l, bend left=25, "j_{!}"] \arrow[r, bend left=25, "i^{*}"] &
C' \arrow[l, bend left=25, "i_{*}"]
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
C \arrow[r, bend left=25, "i^{*}"] & C' \arrow[l, bend left=25, "i_{*}"]
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
C'' \arrow[r, bend left=25, "j_{*}"] &
C \arrow[l, bend left=25, "j^{*}"] \arrow[r, bend left=25, "i^{*}"] &
C' \arrow[l, bend left=25, "i_{!}"]
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
A \arrow[r, bend left=25, "f"] & B \arrow[l, bend left=25, "g"]
\end{tikzcd}

\begin{tikzcd}[column sep=large, row sep=small, nodes={font=\scriptsize}]
\varphi(X) \arrow[r, bend left=18, "\varphi * \beta"]
           \arrow[r, bend right=18, "\beta * \varphi"'] \arrow[d, no head] &
\varphi^{2}(X) \arrow[r, bend left=25] \arrow[r] \arrow[r, bend right=25]
           \arrow[d, no head] &
\varphi^{3}(X) \arrow[d, no head] \\
\Phi^{0}(X) & \Phi^{1}(X) & \Phi^{2}(X)
\end{tikzcd}

\begin{tikzcd}[column sep=large, row sep=small, nodes={font=\scriptsize}]
\varphi \arrow[r, bend left=18] \arrow[r, bend right=18] \arrow[d, no head] &
\varphi^{2} \arrow[r, bend left=25] \arrow[r] \arrow[r, bend right=25]
  \arrow[d, no head] &
\varphi^{3} \arrow[d, no head] \\
\Phi^{0} & \Phi^{1} & \Phi^{2}
\end{tikzcd}

\begin{tikzcd}[column sep=large, row sep=small]
X \arrow[d, "u"'] & \\
\varphi(X) \arrow[r, bend left=18, "\varphi(u)"]
           \arrow[r, bend right=18, "\lambda"'] & \varphi^{2}(X)
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
      (\mathrm{Top}) \arrow[r] &
      (\mathrm{Cat}) \arrow[l, bend left=30, "C \mapsto \widehat{C}"]
      \end{tikzcd}

\begin{tikzcd}[column sep=huge]
      (\mathrm{Ens.}) \arrow[r] &
      (\mathrm{Esp.}) \arrow[r] &
      (\mathrm{Top}) \arrow[ll, bend left=30]
      \end{tikzcd}

\begin{tikzcd}[column sep=huge]
e \arrow[r, bend left=25, "u"] & a \arrow[l, bend left=25, "v"]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
f_{!} & f^{*} \arrow[d, no head] & f_{*} \arrow[d, no head] &
  f^{!} \arrow[d, no head] & f_{?} \arrow[d, no head] \\
& g_{!} & g^{*} \arrow[d, no head] & g_{*} \arrow[d, no head] &
  g^{!} \arrow[d, no head] \\
& & h_{!} & h^{*} & h_{*}
\end{tikzcd}

\begin{tikzcd}
& \varprojlim E/B \arrow[r, hook, "i"] \arrow[d, "F^{*}_{c}"']
  & \underline{\Gamma}(E/B) \arrow[d, "F^{*}"] \\
X' \in B & \varprojlim E'/B' \arrow[r, hook, "i'"]
  & \underline{\Gamma}(E'/B')
\end{tikzcd}