It's quite easy:
\documentclass{article}
\usepackage{amsmath}
\usepackage[all,cmtip]{xy}
\begin{document}
\[
\xymatrix@R+1pc@C+1pc{
E \ar[r]^*{e} & X \ar[r]<4pt>^*{s} \ar[r]<-4pt>_*{t} & Y \\
Z \ar[u]<4pt>^*{\gamma_1} \ar[u]<-4pt>_*{\gamma_2}
\ar[ur]_*{e \circ \gamma_1 = e \circ \gamma_2}
}
\]
\end{document}
With @R+1pc I tell Xy-pic to increase by 1pc (12pt) the interrow space; similarly, @C+1pc increases the intercolumn space. You'd get the same result with
\xymatrix@+1pc{
which would apply the setting to both the interrow and intercolumn spaces. Specifying them separately allows for finer control.

Note you should use \[...\] and not $$, see Why is \[ ... \] preferable to $$ ... $$? You also need no xy environment.
The same with tikz-cd:
\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz-cd}
\begin{document}
\[
\begin{tikzcd}[column sep=3em,row sep=3em]
E \arrow[r,"e"] & X \arrow[r,shift left=.7ex,"s"] \arrow[r,shift right=.7ex,swap,"t"] & Y
\\
Z \arrow[u,shift left=.7ex,"\gamma_1"] \arrow[u,shift right=.7ex,swap,"\gamma_2"]
\arrow[ur,swap,"e \circ \gamma_1 = e \circ \gamma_2"]
\end{tikzcd}
\]
\end{document}
