\begin{tikzcd}
X \arrow[d, "f"'] \arrow[dr, "\Gamma_f"] & \\
Y \arrow[d, "g"'] & X \times_S Y \arrow[l, "\mathrm{pr}_2"'] \arrow[d, "\mathrm{pr}_1"] \\
S & X \arrow[l, "gf"]
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, "\Gamma_f"] \arrow[d, "f"'] & X \times_S Y \arrow[d, "f \times \mathrm{id}_Y"] \\
Y \arrow[r, "\operatorname{diag}_Y"'] & Y \times_S Y
\end{tikzcd}

\begin{tikzcd}
\pi(C, \widetilde{D}) \arrow[r, "{\alpha = \pi(i, j)}"] \arrow[d, "\gamma"'] & \pi(C', \widehat{D}) \approx \widehat{C' \times D} \arrow[d, "\gamma'"] \\
\widehat{C} \times \widetilde{D} \arrow[r, hook, "\beta = i \times j"'] & \widehat{C'} \times \widehat{D}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& \mathfrak{X} \times_S \mathfrak{Y} \arrow[dl] \arrow[dr] & \\
x \in \mathfrak{X} \arrow[dr, no head] & & \mathfrak{Y} \ni y \arrow[dl, no head] \\
& S \ni s &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& K' \text{ corps} \arrow[dl, no head] \arrow[dr, no head] & \\
k(x) \arrow[dr, no head] & & k(y) \arrow[dl, no head] \\
& k(s) &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& Z = V_{K'} \arrow[dl, no head] \arrow[dr, no head] & \\
V_{k(x)} \to V_A & & V_B \leftarrow V_{k(y)} \\
x \to X \arrow[dr] & & Y \leftarrow y \arrow[dl, no head] \\
& S = V_k &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
& C \arrow[dl, no head] \arrow[dr, no head] & \\
A \arrow[dr, no head] & & B \arrow[dl, no head] \\
& k &
\end{tikzcd}

\begin{tikzcd}[column sep=large]
X/F \arrow[d, "\wr" description, no head] & X'/F' \arrow[d, no head] \\
\underline{\operatorname{Hom}}_X(F, \mathbb{L}_X) \arrow[r] & X' \\
X \arrow[u, hook] \arrow[ur] &
\end{tikzcd}

\begin{tikzcd}
X \arrow[d] & X' \arrow[l] \arrow[d] & T \arrow[dl] \arrow[ll, bend right=30] \\
S & S' \arrow[l] &
\end{tikzcd}

\begin{tikzcd}
B \arrow[r] & B' \arrow[r, dashed] & C' \\
A \arrow[u] \arrow[r] & A' \arrow[u] \arrow[ur] &
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, "{(u, v)}"] & Y \times Y \\
K \arrow[u] \arrow[r] & Y \arrow[u, "\Delta_Y"']
\end{tikzcd}

\begin{tikzcd}
B \arrow[d] & A \otimes A \arrow[l, "{(u, v)}"'] \arrow[d] \\
C = ? & A \arrow[l]
\end{tikzcd}

\begin{tikzcd}
G \arrow[r] & X \\
G_{(x)} \arrow[u] \arrow[r] & e_k \arrow[u]
\end{tikzcd}

\begin{tikzcd}
& M \arrow[dl, "i_g"'] \arrow[d, "i_f"] \\
M_g \arrow[r, "i_{f,g}"'] & M_f
\end{tikzcd}