Unusual diagram of snake lemma

Question

I want to draw the following diagram in LaTeX. This is essentially a diagram of the snake lemma, but a bit different from the usual one.

By rotating the diagram of the snake lemma ninety degrees and inverting it, this diagram can be obtained. But, to write this directly may be more comfortable.

Thank you for your help.

The following is the code for the usual snake lemma.

\begin{document}

\documentclass{article}
\thispagestyle{empty}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{tikz-cd}
\usetikzlibrary{%
  matrix,%
  calc,%
  arrows%
}

\begin{tikzpicture}[>=triangle 60]

\matrix[matrix of math nodes,column sep={60pt,between origins},row
sep={60pt,between origins},nodes={asymmetrical rectangle}] (s)
{
&|[name=ka]| \ker f &|[name=kb]| \ker g &|[name=kc]| \ker h \\
%
&|[name=A]| A' &|[name=B]| B' &|[name=C]| C' &|[name=01]| 0 \\
%
|[name=02]| 0 &|[name=A']| A &|[name=B']| B &|[name=C']| C \\
%
&|[name=ca]| \coker f &|[name=cb]| \coker g &|[name=cc]| \coker h \\
};
\draw[->] (ka) edge (A)
          (kb) edge (B)
          (kc) edge (C)
          (A) edge (B)
          (B) edge node[auto] {\(p\)} (C)
          (C) edge (01)
          (A) edge node[auto] {\(f\)} (A')
          (B) edge node[auto] {\(g\)} (B')
          (C) edge node[auto] {\(h\)} (C')
          (02) edge (A')
          (A') edge node[auto] {\(i\)} (B')
          (B') edge (C')
          (A') edge (ca)
          (B') edge (cb)
          (C') edge (cc)
;
\draw[->,gray] (ka) edge (kb)
               (kb) edge (kc)
               (ca) edge (cb)
               (cb) edge (cc)
;
\draw[->,gray,rounded corners] (kc) -| node[auto,text=black,pos=.7]
{\(\partial\)} ($(01.east)+(.5,0)$) |- ($(B)!.35!(B')$) -|
($(02.west)+(-.5,0)$) |- (ca);
\end{tikzpicture}

The output is

The code for this usual snake lemma diagram can be also seen in How do you draw the "snake" arrow for the connecting homomorphism in the snake lemma?

Answer 1

You can bend the arrow:

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{cd,bbox}

\DeclareMathOperator{\Coker}{Coker}

\begin{document}

Some text before the diagram, we make it long enough to split across
lines so we see better what happens.
\[
\begin{tikzcd}[
  column sep={5em,between origins},
  row sep=5ex,
  bezier bounding box, % <-- https://tex.stackexchange.com/a/619994/
]
0 \arrow[r] &
A \arrow[r] \arrow[d] &
B \arrow[r,"i"] \arrow[d] &
C \arrow[r] \arrow[d] &
\Coker i \arrow[r] \arrow[d,"g"] &
0
\\
0 \arrow[r] &
D \arrow[r] \arrow[d] &
E \arrow[r,"i"] \arrow[d] &
F \arrow[r] \arrow[d] &
J
\\
&
G \arrow[r] \arrow[rrruu,in=150,out=170,looseness=2.6,"\delta"] &
H \arrow[r] &
I
\end{tikzcd}
\]
Some text after the diagram, we make it long enough to split across
lines so we see better what happens.

\end{document}

Answer 2

With tikz-cd you can control the arrow using to path in the arrow options. Here is the original together with your version for comparison.

\documentclass{article}

\usepackage{tikz-cd, amsmath}

\DeclareMathOperator{\coker}{coker}
\tikzcdset{arrow style=tikz, diagrams={>={Triangle[]}}}
\colorlet{mygray}{gray!80}

\begin{document}

\[
\begin{tikzcd}[sep={60pt,between origins}, arrows=thick, every label/.append style={font=\normalsize}]
 & \ker f\arrow[d]\arrow[r, mygray] & \ker g\arrow[d]\arrow[r, mygray] & \ker h\arrow[d] & \\ % Final `&` needed for proper spacing in subsequent rows
 & A'\arrow[d, "f"]\arrow[r] & B'\arrow[d, "g", ""{coordinate, name=Z, pos=.2}]\arrow[r, "p"] & C'\arrow[d, "h"]\arrow[r] & 0\\
0\arrow[r] & A\arrow[d]\arrow[r, "i"] & B\arrow[d]\arrow[r] & C\arrow[d]\\
 & \coker f\arrow[r, mygray] & \coker g\arrow[r, mygray] & \coker h
\arrow[mygray, from=1-4, to=4-2, rounded corners, 
    to path={-- ([xshift=60pt]\tikztostart.east) |- (Z) [pos=.25, text=black]\tikztonodes 
    -| ([xshift=-60pt]\tikztotarget.west) -- (\tikztotarget)}, "\partial"]
\end{tikzcd}
\]

\[
\begin{tikzcd}[sep={60pt,between origins}, arrows=thick, every label/.append style={font=\normalsize}]
0\arrow[r] & A\arrow[r]\arrow[d] & B\arrow[r, "i"]\arrow[d] & C\arrow[r]\arrow[d] & \coker i\arrow[d]\arrow[r] & 0\\
0\arrow[r] & D\arrow[r]\arrow[d] & E\arrow[r]\arrow[d] & F\arrow[r]\arrow[d] & J\\
 & G\arrow[r] & H\arrow[r] & I
\arrow[mygray, from=3-2, to=1-5, rounded corners,
    to path={-- ([xshift=-70pt]\tikztostart.west) --++(0,140pt) [pos=1, text=black]\tikztonodes 
    -| (\tikztotarget.north) }, "\partial"]
\end{tikzcd}
\]

\end{document}