Vertical cofibration arrow in xypic

As you told, from the user guide, we have \newdir{ >}{{}*!/-7pt/\dir{>}}.

Note the empty space before the first use of >.

(Edited: I changed the value to -7pt. Choose what is better to you.)

Then you use it as \ar@{ >->}.

See the result:

a one-line change in your existing example, adding cmtip to

\usepackage[all,cmtip]{xy}

gives this result:

If you are willing to switch to tikz-cd, here's the same diagram:

\documentclass{article}
\usepackage{tikz-cd}

\begin{document}

\[
\begin{tikzcd}
S_{P} \arrow[equal]{r} \arrow[tail]{d} &
 S_{P} \arrow[two heads]{r} \arrow[tail]{d} &
 0 \arrow[tail]{d} \\
R' \arrow[tail]{r} \arrow[two heads]{d} &
 R \arrow[two heads]{r} \arrow[two heads]{d} &
 M \arrow[equal]{d} \\
P' \arrow[tail]{r} & P \arrow[two heads]{r} & M
\end{tikzcd}
\]

\end{document}