\documentclass{ppex}
\nonstopmode
\usepackage [arrow,matrix,tips]{xy}
\pagestyle{empty}
\setlength\textwidth{207.0pt}
\begin{document}
\ResetPreambleCommands
\ReadyForTheFray
\usepackage [arrow,matrix,tips]{xy}
\[\UseTips
\newdir{ >}{!/-5pt/\dir{>}}
\xymatrix @=1pc @*[r] @*[o] {
  & \Sigma^L \ar[rrrr]^{\varphi^r}
             \ar'[d]'[ddd]_{\varphi^m}[dddd]
  & & & & \Sigma^R \ar[dddd]^{\varphi^{m^*}}
\\ L   \ar[ur]^{\lambda^L}
  & & L_r \ar@{ >->}[ll]_(.32){i_1} \ar[rr]^r
  & & R \ar[ur]_{\lambda^R}
\\ \\
   L_m \ar@{ >->}[uu]^{i_2} \ar[dd]_m
  & & K_{r,m} \ar@{ >->}[ll]_(.3){i_3}
              \ar@{ >->}[uu]_{i_4}
              \ar[dd]_<<<<m \ar[rr]^r
  & & R_{m^*} \ar@{ >->}[uu]_{i_6}
              \ar[dd]_<<<<{m^*}
\\  &  \Sigma^G
         \ar'[r]'[rrr]_{\varphi^{r^*}}[rrrr]
  & & & & \Sigma^H
\\ G \ar[ur]^(.7){\lambda^G}
  & & G_{r^*} \ar@{ >->}[ll]^{i_5}
              \ar[rr]_{r^*}
  & & H \ar@{.>}[ur]_{\lambda^H}
}\]
\end{document}