\begin{tikzcd}
\underline{C}_S \arrow[r] \arrow[d] & \underline{C}_T \arrow[d] \\
\underline{Et}_S \arrow[r, "f^{\circ}"] & \underline{Et}_T
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
\underline{C}_S \arrow[r] \arrow[d, "\underline{s}_S"'] & \underline{C}_T \arrow[d, "\underline{s}_T"] \\
(\mathrm{Sch})_{/S} \arrow[r, "X \mapsto X \times_S T"'] & (\mathrm{Sch})_{/T}
\end{tikzcd}

\begin{tikzcd}[column sep=huge]
\mathrm{Rev}^{\underline{R}_{(S)}}(S) \arrow[r] \arrow[d] & \mathrm{Rev}^{\underline{R}_{(T)}}(T) \arrow[d] \\
\mathrm{Rev}(S) \arrow[r, "X \mapsto X \times_S T"'] & \mathrm{Rev}(T)
\end{tikzcd}

\begin{tikzcd}
\hat{A}^{(x)} & & \hat{A}^{(y)} \\
& A \arrow[ul] \arrow[ur] &
\end{tikzcd}

\begin{tikzcd}
\mathrm{Coh}(\mathfrak{X}) & \mathrm{Coh}(X) \arrow[l, "\approx"'] \\
\mathrm{Coh}(\mathfrak{X})_{\mathrm{nilp}} \arrow[u] & \mathrm{Coh}(X)_{\mathrm{nilp}} \arrow[u]
\end{tikzcd}

\begin{tikzcd}
X \arrow[d] & X_1 \arrow[l] \arrow[d] \\
Y & Y_1 \arrow[l]
\end{tikzcd}

\begin{tikzcd}
X' \arrow[d, "p"] & \supset & D'_i \arrow[d] \\
X & \supset & D
\end{tikzcd}

\begin{tikzcd}
C \arrow[r] \arrow[d, hook'] & X \arrow[d, hook'] \\
V \arrow[r] & U
\end{tikzcd}

\begin{tikzcd}
C \times Y = C_Y \arrow[r] \arrow[d, hook'] & R \arrow[d, hook'] \\
V \times Y = V_Y \arrow[r] & P
\end{tikzcd}

\begin{tikzcd}
X' \arrow[d] \\
X
\end{tikzcd}

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

\begin{tikzcd}
X'' \arrow[d, Rightarrow] \\
X' \arrow[d] \\
X \arrow[d] \\
{}
\end{tikzcd}

\begin{tikzcd}
Y' \arrow[r] \arrow[d] & X' \arrow[d] \\
Y \arrow[r, hook] \arrow[d] & X \arrow[d] \\
{} & \mathbf{P}^r
\end{tikzcd}

\begin{tikzcd}
X' \arrow[r, hook] \arrow[d] & \bar X' \arrow[d] \\
X \arrow[r, hook] \arrow[d] & \bar X \arrow[dl] \\
\mathbf{P}^r
\end{tikzcd}

\begin{tikzcd}
Y \arrow[r] \arrow[d] & X \arrow[d] \\
H \arrow[r] & \mathbf{P}^r
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X'_u \arrow[r] \arrow[d] & X' \arrow[d] \\
X_u \arrow[r] \arrow[d] & X_K \arrow[r] \arrow[d] & X \arrow[d] \\
L_u \arrow[r] & \mathbf{P}^r_K \arrow[r] & \mathbf{P}^r_k
\end{tikzcd}

\begin{tikzcd}
& X'' \arrow[dl] \arrow[dr] & \\
X & & X'
\end{tikzcd}

\begin{tikzcd}
X_1 \arrow[d, "f"] & X_1 \arrow[l] \\
X_0 & \supset U_1 \supset U_0
\end{tikzcd}

\begin{tikzcd}
\pi_1(X_1) \arrow[d, "f_{*}"'] & \pi_1(U_1) \arrow[l] & \\
\pi_1(X_0) & \pi_1(U_0) \arrow[l] \arrow[r, "\sim"] \arrow[ul, "g_{*}"] & \pi_1(U_1)
\end{tikzcd}

\begin{tikzcd}
& X \arrow[d, "f"] \\
U \arrow[ur, "g"] \arrow[r, "i"'] & Y
\end{tikzcd}

\begin{tikzcd}
& \mathrm{Pic}(X) \arrow[dl, "g^{*}"] \\
\mathrm{Pic}(U) & \mathrm{Pic}(Y) \arrow[u, "f^{*}"'] \arrow[l, "i^{*}"]
\end{tikzcd}

\begin{tikzcd}
& T(X) \arrow[dl, "g^{*}"] \\
T(U) & T(Y) \arrow[u, "f^{*}"'] \arrow[l, "i^{*}"]
\end{tikzcd}

\begin{tikzcd}
C' \arrow[r, hook] \arrow[d] & X' \arrow[d] \\
C_{k(u)} \arrow[r, hook] & X \arrow[r, hook] & \mathbf{P}^r
\end{tikzcd}

\begin{tikzcd}
X \times_{S} S' \arrow[r] \arrow[d] & X \arrow[d] \\
S' \arrow[r] & S
\end{tikzcd}

\begin{tikzcd}
I_{o}(\struck{\ill{}}) \arrow[r] \arrow[d] & I'_{o} \arrow[d] \\
I(\struck{\ill{}}) \arrow[r] & I'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \hat{Z} = \pi_{1}(Z') \arrow[dl] \arrow[dr] & \pi_{1}(Z'') = \hat{Z}
 \arrow[dl] \arrow[d] \\
\pi_{1}(X_{T'}) \arrow[dr] & \pi_{1}(X_{T''}) \arrow[d] & \pi_{1}(Z) =
\hat{Z}^{2} \arrow[dl] \\
 & \pi_{1}(X_{T}) &
\end{tikzcd}

\begin{tikzcd}
S & \hat{S} \arrow[l] \\
X \arrow[u] & \hat{X} \arrow[l] \arrow[u]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
G_{\xi} \arrow[r, "\phi"] & H_{y} \\
\prod_{\ell} \underline{Z}_{\ell}(1)(\bar{x})^{C} \arrow[u] \arrow[r, "N"] &
\prod_{\ell} \underline{Z}_{\ell}(1)(\bar{y})^{C'} \arrow[u]
\end{tikzcd}

\begin{tikzcd}
X \arrow[d, "f"] \\
S
\end{tikzcd}

\begin{tikzcd}
X(s) \arrow[d] & X(t) \arrow[l, "\varphi_{st}"'] \arrow[d] \\
\operatorname{Spec}(\mathcal{O}_{s}) & \operatorname{Spec}(\mathcal{O}_{t}) \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X(s) \arrow[d] & X \arrow[l] \arrow[d] & X \times_{U} \operatorname{Spec}(\mathcal{O}_{t}) \arrow[l] \arrow[d] \arrow[r, dashed] & X(t) \\
\operatorname{Spec}(\mathcal{O}_{s}) \arrow[r] & U & \operatorname{Spec}(\mathcal{O}_{t}) \arrow[l] &
\end{tikzcd}

\begin{tikzcd}
(P, \xi) \arrow[r] & (Q, \eta) \\
& (P', \xi') \arrow[u]
\end{tikzcd}

\begin{tikzcd}
G \arrow[r, "{(u,v)}"] & G' \times G' \\
& G' \arrow[u]
\end{tikzcd}

\begin{tikzcd}
G \arrow[r, "{(u,v)}"] & G' \times G' \\
H \arrow[u] \arrow[r] & G' \arrow[u]
\end{tikzcd}

\begin{tikzcd}
(P, \xi) \arrow[r] & (Q, \eta) \\
(R, \lambda) \arrow[u] \arrow[r] & (P', \xi') \arrow[u]
\end{tikzcd}

\begin{tikzcd}[column sep=large]
P \arrow[r, "\mathrm{can}"] \arrow[rr, bend right=30, "\mathrm{can}"'] & P \times_u G' \arrow[r, "w"] & P \times_v G'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
(p_1 p_{12})^{-1}(E) \arrow[r] \arrow[d] & (p_2 p_{12})^{-1}(E) = (p_1 p_{23})^{-1}(E) \arrow[d] \\
(p_1 p_{13})^{-1}(E) \arrow[r] & (p_2 p_{13})^{-1}(E) = (p_2 p_{23})^{-1}(E)
\end{tikzcd}

\begin{tikzcd}[column sep=large]
E \arrow[r, "A_{\alpha'}"] \arrow[d, "a'_{\beta}"'] & E \arrow[r, "a''_{\beta}", "\sim"'] & E \arrow[d, "A_{\alpha''}"] \\
E \arrow[r, "A_{\alpha'''}"'] & E \arrow[r, "a'''_{\beta}"', "\sim"] & E
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& R_0 \arrow[d] & R_1 \arrow[d] & R_2 \arrow[d] \\
S & S' \arrow[l] & S'' \arrow[l] & S''' \arrow[l]
\end{tikzcd}

\begin{tikzcd}[row sep=small, column sep=large]
X \arrow[d] & \\
S & S'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\pi_1(X \times_S Y) & & \\
\pi_1(X) \arrow[d, "\mathrm{surj}"'] & \pi_1(Y) \arrow[dl, "\mathrm{surj}"] & \pi_1(X') \arrow[d] \\
\pi_1(S) \arrow[rr] & & \pi_1(S')
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& \pi_1(X) \times_{\pi_1(S)} \pi_1(Y) \arrow[dl] \arrow[dr] & \\
\pi_1(X) \arrow[dr, "\mathrm{surj}"'] & & \pi_1(Y) \arrow[dl] \\
& \pi_1(S) &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& \operatorname{Im}(\pi_1(\bar{X}_s) \to \pi_1(X)) \arrow[d, no head] & & & \\
e \arrow[r] & \operatorname{Ker}[\pi_1(X) \to \pi_1(S)] \arrow[r, "\sim"] & \pi_1(X) \times_{\pi_1(S)} \pi_1(Y) \arrow[r] & \pi_1(Y) \arrow[r] & e \\
e \arrow[r] & \operatorname{Ker}[\pi_1(X \times_S Y) \to \pi_1(Y)] \arrow[u] \arrow[d, no head] \arrow[r] & \pi_1(X \times_S Y) \arrow[u] \arrow[r] & \pi_1(Y) \arrow[u] \arrow[r] & e \\
& \operatorname{Im}(\pi_1(\bar{X}_s) \to \pi_1(X \times_S Y)) & & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\bar{X}_s \arrow[r] \arrow[d, no head] & X \times_S Y \arrow[d] \\
\bar{X}_s \arrow[r] & X \arrow[d] \\
& S
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\pi_1(\bar{X}_s) \arrow[r] \arrow[d, no head] & \pi_1(X \times_S Y) \arrow[d] \\
\pi_1(\bar{X}_s) \arrow[r] & \pi_1(X) \arrow[d] \\
& \pi_1(S)
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[dr] & \\
& X' = X/G \arrow[d] \\
& Y \ni y
\end{tikzcd}

\begin{tikzcd}[row sep=small]
k(x') \arrow[d, no head] \\
k(x) \arrow[d, no head] \\
k(y)
\end{tikzcd}

\begin{tikzcd}
\xi \arrow[r, "x"] \arrow[d] & X \arrow[d] \\
\xi' \arrow[r, "x'"'] & X'
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& X \arrow[d] \\
S & T \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\widetilde{\widetilde{X}}{}^i \arrow[d] & \\
\widetilde{X}^i \arrow[d] & \widetilde{X} \arrow[d, no head] \\
X^i \arrow[r] & X
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X \arrow[d] & X' \arrow[l] \arrow[d] & X'' \arrow[l, bend left=12]
  \arrow[l, bend right=12] \arrow[d] \\
Y & Y' \arrow[l] & Y'' \arrow[l, bend left=12] \arrow[l, bend right=12]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\pi_1(Y) & \pi_1(Y') \arrow[l] & \pi_1(Y'') \arrow[l, bend left=12]
  \arrow[l, bend right=12] \arrow[d] & \pi_1(Y''') \arrow[l, bend left=18]
  \arrow[l] \arrow[l, bend right=18] \\
& & \mathfrak{z} &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
Y & X_0^0 \arrow[l, dashed] & X_1^1 \arrow[l, bend left=12]
  \arrow[l, bend right=12] & X_2^2 \arrow[l, bend left=20] \arrow[l]
  \arrow[l, bend right=20] \\
\xi \arrow[u] & X_0^{\xi} \arrow[l] \arrow[u] & X_1^{\xi}
  \arrow[l, bend left=12] \arrow[l, bend right=12] \arrow[u] & X_2^{\xi}
  \arrow[l, bend left=20] \arrow[l] \arrow[l, bend right=20] \arrow[u] \\
& K_0^0 & K_1^1 \arrow[l, bend left=12] \arrow[l, bend right=12] & K_2^2
  \arrow[l, bend left=20] \arrow[l] \arrow[l, bend right=20] \\
& k_0^0 \arrow[u, "i_0"] & k_1^1 \arrow[l] \arrow[u, "i_1"] & k_2^2
  \arrow[l, bend left=12] \arrow[l, bend right=12] \arrow[u, "i_2"]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\overline{K}_0 & \overline{K}_1 \arrow[l, bend left=12] \arrow[l, bend right=12]
  & \overline{K}_2 \arrow[l, bend left=20] \arrow[l] \arrow[l, bend right=20]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
F^0 \arrow[r, bend left=12] \arrow[r, bend right=12]
  & F^1 \arrow[r, bend left=20] \arrow[r] \arrow[r, bend right=20] & F^2
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\overline{K}_0 & \overline{K}_1 \arrow[l, bend left=12] \arrow[l, bend right=12]
  & \overline{K}_2 \arrow[l, bend left=20] \arrow[l] \arrow[l, bend right=20]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
K_0 & K_1 \arrow[l, bend left=12] \arrow[l, bend right=12]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
F^0 \arrow[r, bend left=12] \arrow[r, bend right=12]
  & F^1 \arrow[r, bend left=20] \arrow[r] \arrow[r, bend right=20] & F^2
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
F^0 \arrow[r, bend left=12] \arrow[r, bend right=12] \arrow[d, "\xi^0"]
  & F^1 \arrow[r, bend left=20] \arrow[r] \arrow[r, bend right=20]
    \arrow[d, "\xi^1"] & F^2 \arrow[d, "\xi^2"] \\
f^0 \arrow[r, bend left=12] \arrow[r, bend right=12]
  & f^1 \arrow[r, bend left=20] \arrow[r] \arrow[r, bend right=20] & f^2
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
K_0 & K_1 \arrow[l, bend left=12] \arrow[l, bend right=12]
  & K_2 \arrow[l, bend left=20] \arrow[l] \arrow[l, bend right=20]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
k_0 & k_1 \arrow[l, bend left=12] \arrow[l, bend right=12]
  & k_2 \arrow[l, bend left=20] \arrow[l] \arrow[l, bend right=20]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& X'' \arrow[dl, "\text{fini}"'] \arrow[d] \\
X \arrow[d] & X' \\
Y &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
X_i & X'_i \arrow[l] \\
X \arrow[u] & X' \arrow[l] \arrow[u]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
X_i \arrow[d] & X_{i\alpha} \arrow[d] \\
X & X_{\alpha} \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& \mathrm{Rev}\,\mathrm{Fl}_S \arrow[d] \\
\mathcal{G} \arrow[r] \arrow[d] & \mathrm{Fl}_S \\
\mathrm{Et}(S) &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
T \arrow[d, bend left=12] \arrow[d, bend right=12]
  & X \times T \arrow[d, bend left=12] \arrow[d, bend right=12] & \\
\varphi(M) \arrow[d]
  & X \times \varphi(M) \arrow[l, no head] \arrow[r] \arrow[d] & X_1 \\
e & X \arrow[l, no head] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& \varphi(M) \times_{\varphi} M' \arrow[dl] \arrow[dr] & \\
\varphi(M) & & \varphi(M')
\end{tikzcd}

\begin{tikzcd}[column sep=large]
X^P \arrow[r, hook, "\alpha_P"] & X^E
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
X(1) \arrow[dr] & & X(2) \arrow[dl] \\
& y &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[d] & X \times_Y X = X' \arrow[l] \arrow[d] \\
Y & X = Y' \arrow[l, no head]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
Z'' \arrow[d]
  & Z \arrow[l, dashed] \arrow[d]
  & D \arrow[l, bend left=12] \arrow[l, bend right=12] \arrow[d] \\
Y & X \arrow[l]
  & X \times_Y X \arrow[l, bend left=12] \arrow[l, bend right=12]
\end{tikzcd}