\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\alpha \arrow[r, two heads] \arrow[d, "\xi"'] & \beta \arrow[d, dashed] & \gamma \arrow[l, hook'] \arrow[d, "\eta"] \\
X \arrow[r, two heads] & Y & Z \arrow[l, hook']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\alpha_0 \arrow[r, two heads] \arrow[d, "\xi_0"'] & \alpha'_0 \arrow[d, dashed, "\xi'_0"'] & \alpha_1 \arrow[l, hook'] \arrow[r] \arrow[d, "\xi_1"'] & \alpha_2 \arrow[r] \arrow[d, "\xi_2"'] & \cdots \arrow[r] & \alpha_{n-1} \arrow[r, two heads] \arrow[d, "\xi_{n-1}"'] & \alpha'_{n-1} \arrow[d, dashed, "\xi'_{n-1}"'] & \alpha_n \arrow[l, hook'] \arrow[d, "\xi_n"] \\
X_0 \arrow[r, two heads] & X'_0 & X_1 \arrow[l, hook'] \arrow[r] & X_2 \arrow[r] & \cdots \arrow[r] & X_{n-1} \arrow[r, two heads] & X'_{n-1} & X_n \arrow[l, hook']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
(\mathrm{Add}_k) \arrow[r, "2\text{-ess. surj.}"] \arrow[d, "\operatorname{Kar}"'] & (\mathrm{Top\ ab}\ \ill{}_{\,k}) \\
(\mathrm{AddKar}_k) \arrow[ur, "3\text{-fid.}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
M \arrow[r, "\text{pl. fid.}"] & M^{k} \\
\operatorname{Hom}_k(P^{\circ}, M) \arrow[r, "\text{pl. fid.}"] & \operatorname{Hom}_k(P^{\circ}, M^{k}) \arrow[d, "\text{pl. fid.}"] \\
& \operatorname{Hom}_k(P^{k\circ}, M^{k}) \\
& \operatorname{Bil}_k(P^{k} \times M^{\circ}, Ab_k)
\end{tikzcd}