xypic, xymatrix, aligning entries in a column

Question

I am working with the following code:

\documentclass[12pt,letterpaper]{article}
\usepackage[all]{xy}
\begin{document}

\[
\xymatrix
{
                &   A \ar[dd]_{\pi_A}                                   \\
                &   &   **[r] G(A) = G(B)=Z\oplus Z\oplus Z\oplus Z     \\
**[l] 4\beta+ f(\beta) + g(\beta) = &   B \ar[dd]_{\pi_B}               \\
                &   &   **[r] F(B) = F(C)                               \\
**[l] \gamma=   &   C
}
\]
\end{document}

This code creates the following diagram:

It looks pretty good, except that I would like the left-most column to be much closer, almost touching the center column, and also for the right-most column to be much closer, but maybe an inch or so away from the center column.

If I change the line \xymatrix to \xymatrix@C=-45pt, I get something pretty good, but this solution seems very ad-hoc to me. I think that the issue is that the column alignment is keeping the centerline of the column the same, and keeping the column widths the same, and aligning the entries to flush right (or left) along this centerline, rather than flush right (or left) in the space allotted to the column.

Answer 1

Would this be better?

\[
\hphantom{4\beta+ f(\beta) + g(\beta) ={}}
\xymatrix@R+1pc
{
A \ar[d]_{\pi_A}^{\qquad\textstyle G(A) = G(B)=Z\oplus Z\oplus Z\oplus Z} \\
\llap{$4\beta+ f(\beta) + g(\beta) ={}$}B \ar[d]_{\pi_B}^{\qquad\textstyle F(B) = F(C)} \\
\llap{$\gamma={}$}C
}
\]