The MnSymbol
package offers \upmapsto
and \downmapsto
:
\documentclass{article}
\usepackage{MnSymbol}
\begin{document}
$\upmapsto\quad\downmapsto$
\end{document}

However, using this package will change some other symbols, and this might be no desirable. In this particular case, since the arrow is needed for a commutative diagram I suggest using the tikz-cd
package and its mapsto
option for arrows:
\documentclass{article}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A \arrow{r}{\psi} \arrow[mapsto,color=red]{d}
& B \\
C \arrow[mapsto,color=red]{r}[color=blue]{\eta}
& D \arrow[mapsto,color=red]{u}
\end{tikzcd}
\end{document}

Now that an edit has been made to the original question including the desired commutative diagram, here it is using tikz-cd
:
\documentclass{article}
\usepackage{tikz-cd}
\newcommand\tikzmark[5][0]{%
\tikz[overlay,remember picture,baseline] \node [rotate=#1,anchor=base,xshift=#4,yshift=#5] (#2) {$\scriptstyle#3$};}
\begin{document}
\begin{tikzcd}
0 \arrow{r} &
A_{n+1} \arrow{r} \arrow{d} &
B_{n+1} \arrow{r} \arrow{d} &
C_{n+1} \arrow{r} \arrow{d} &
0 \arrow{d}
\\
&
A_n \arrow{r} \arrow{d} &
\tikzmark{s}{b}{14pt}{-9pt}B_n \arrow{r} \arrow{d}{d_n} &
\tikzmark{s}{c}{-2pt}{-8pt}\tikzmark{e}{0}{-2pt}{-24pt}\tikzmark[270]{a}{\mapsto}{14pt}{-4pt}
C_n \arrow{r}{g_n} \arrow{d} &
0 \arrow{d}
\\
&
A_{n-1} \arrow{r}{a} &
B_{n-1} \arrow{r}{f_{n-1}} &
C_{n-1} \arrow{r} & 0
\end{tikzcd}
\end{document}
A variation of \tikzmark
was used to position some special characters; in particular, the vertical "downmapsto" arrow was obtained with the rotate=270
option applied to a standard \mapsto

Best Answer
You can get automatically the § symbol (which I don't recommend, though) with
cleveref
and its\cref
command.It's better to have a different command than
\ref
, because you might need the latter for referring the section without the § prefix. The § symbol is obtained with\S
.You can also format multiple references; here's a starting point.