Commutative diagram with parallel arrows in opposite direction

First of all don't use align* for a single object and prefer equation*.

Simple commutative diagrams with the CD environment can have arrows pointing

• right

  @>>>
  

• left

  @<<<
  

• down

  @VVV
  

• up

  @AAA
  

• double stem, no arrow head

  @=
  

• nothing (for filling blanks)

  @.
  

However CD doesn't allow for "double arrows" and you have to go to a more specialized package. For instance xy:

\documentclass{article}
\usepackage{amsmath}
\usepackage[all,cmtip]{xy}
\begin{document}
\begin{equation*}
\xymatrix{
   W \ar[r]^{\phi} \ar@<-2pt>[d]_{a} & X \ar@<-2pt>[d]_{c} \\
   Y\ar@<-2pt>[u]_{b} \ar[r]^{\phi} & Z \ar@<-2pt>[u]_{d}
}
\end{equation*}
\end{document}

In addition to egregs answer, one could use the package tikz-cd for this:

% arara: pdflatex

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

\begin{document}     
\[
\begin{tikzcd}
W \arrow{r}{\phi} \arrow[shift right, swap]{d}{a}& X \arrow[shift right, swap]{d}{c}\\
Y \arrow{r}{\phi} \arrow[shift right, swap]{u}{b} & Z \arrow[shift right, swap]{u}{d}
\end{tikzcd}
\]
\end{document}