Snake-lemma and making the connective morphism pass under the others

Question

i'm trying to draw the snake-lemma diagram but the connective morphism is no passing under the other ones as i would like to (it IS passing under, but not with enough spacing, the second image should clarify what i mean) and i would also like to move its label, like the second figure.

\[\begin{tikzcd}
{} & \red{\ker f'} \arrow[r, "\overline{\alpha}", red] \arrow[d, hook, "i'"] & \red{\ker f} \arrow[r, "\overline{\beta}", red] \arrow[d, hook, "i"] & \red{\ker f''} \arrow[d, hook, "i''"] 
\arrow[ddd, phantom, ""{coordinate, name=Z}] \arrow[dddll,
    "\delta",
    rounded corners,
    to path={ -- ([xshift=2ex]\tikztostart.east)
              |- (Z) [near end]\tikztonodes
              -| ([xshift=-2ex]\tikztotarget.west)
              -- (\tikztotarget)}, red] & {}\\
{} & M' \arrow[r, "\alpha"] \arrow[d, "f'"] & M \arrow[r, "\beta"] \arrow[d, "f"] & M'' \arrow[r] \arrow[d, "f''"] & 0 \\
0 \arrow[r] & N' \arrow[r, "\alpha'"] \arrow[d, two heads, "\pi'"] & N \arrow[r, "\beta'"] \arrow[d, two heads, "\pi"] & N'' \arrow[d, two heads, "\pi''"] & {} \\
{} & \red{\coker f'} \arrow[r, "\overline{\alpha'}", red] & \red{\coker f} \arrow[r, "\overline{\beta'}", red] & \red{\coker f''} & {}
\end{tikzcd}\]

Answer 1

You need to find a way to tell tikzcd that f, f', f'' come with white backgrounds. For instance \def\atfield[#1]{\arrow[#1,nodes={fill=white}]}

You may also use preaction={draw=white,-,line width=3} to create white background for arrows.

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz-cd}
\begin{document}
\def\red#1{|[red]|#1}
\def\coker{\operatorname{coker}}
\def\atfield[#1]{\arrow[#1,preaction={draw=white,-,line width=3},nodes={fill=white}]}
[\begin{tikzcd}
{} & \red{\ker f'} \arrow[r, "\overline{\alpha}", red] \arrow[d, hook, "i'"] & \red{\ker f} \arrow[r, "\overline{\beta}", red] \arrow[d, hook, "i"] & \red{\ker f''} \arrow[d, hook, "i''"] 
\arrow[ddd, phantom, ""{coordinate, name=Z}] \arrow[dddll,
    "\qquad\delta",
    rounded corners,
    to path={ -- ([xshift=2ex]\tikztostart.east)
              |- (Z) [near end]\tikztonodes
              -| ([xshift=-2ex]\tikztotarget.west)
              -- (\tikztotarget)}, red] & {}\
{} & M' \arrow[r, "\alpha"] \atfield[d, "f'"] & M \arrow[r, "\beta"] \atfield[d, "f"] & M'' \arrow[r] \atfield[d, "f''"] & 0 \
0 \arrow[r] & N' \arrow[r, "\alpha'"] \arrow[d, two heads, "\pi'"] & N \arrow[r, "\beta'"] \arrow[d, two heads, "\pi"] & N'' \arrow[d, two heads, "\pi''"] & {} \
{} & \red{\coker f'} \arrow[r, "\overline{\alpha'}", red] & \red{\coker f} \arrow[r, "\overline{\beta'}", red] & \red{\coker f''} & {}
\end{tikzcd}]
\end{document}

Answer 2

This doesn't use tikz-cd but is based on the answer given at How do you draw the "snake" arrow for the connecting homomorphism in the snake lemma? . The sizes of the gaps are partly hard coded as they have to encompass both the node and the edge. I haven't adapted it to your labels, but that should be straight forward (I recommend using the asymmetrical rectangle node so that the alpha and beta labels are aligned horizontally).

\documentclass[border=10]{standalone}
%\url{https://tex.stackexchange.com/q/611629/86}
\thispagestyle{empty}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{tikz-cd}
\usetikzlibrary{%
  matrix,%
  calc,%
  arrows%
}
\DeclareMathOperator{\coker}{coker}
\begin{document}
\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,asymmetrical rectangle] (f) {(f)} (A')
          (B) edge node[auto,asymmetrical rectangle] (g) {(g)} (B')
          (C) edge node[auto,asymmetrical rectangle] (h) {(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)} ($(01.east)+(.5,0)$)  |-
(h.mid east) (h.mid west) +(-.1,0) --
(g.mid east) (g.mid west) +(-.1,0) --
(f.mid east) (f.mid west) +(-.1,0)
 -| ($(02.west)+(-.5,0)$) |- (ca);
\end{tikzpicture}
\end{document}