# reshape a diagram with xymatrix

I am using xymatrix for the following diagram

\eq
&&\xymatrix{
&X\times Y\times X\times Y&& \\
&X\times X\times Y \ar@{^{(}->}[u]_-{j}\ar[ld]_{q}\ar[rd]&&\\
X\times X \ar[rd]&&X\times Y\ar[ld]\ar[rd]&\\
&X&&Y
}\eneq

But the central commutative diagram looks like a diamond rather than a square. Is it possible to control better the shape of this. Also, the central diagram is cartesian, so that I would like to draw inside the diagram a rotated square (\pi/4 rotation) with xymatrix (see also here)

Adjust the row and column separation. For the square in the middle, add a phantom arrow.

\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage[all,cmtip]{xy}

\begin{document}

$$\begin{gathered} \xymatrix@C-1.2pc@R+1.6pc{ &X\times Y\times X\times Y && \\ &X\times X\times Y \ar@{^{(}->}[u]_-{j}\ar[ld]_{q}\ar[rd] \ar@{}[dd]|\square&&\\ X\times X \ar[rd]&&X\times Y\ar[ld]\ar[rrd]&\\ &X&&&Y } \end{gathered}$$

\end{document}

With tikz-cd:

\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage{tikz-cd}

\newcommand{\commsquare}[1][0]{%
\raisebox{\depth}{\rotatebox[origin=center]{#1}{$\scriptstyle\square$}}%
}

\begin{document}

$$\begin{tikzcd}[row sep=2.5em,column sep={4em,between origins},nodes in empty cells] & X\times Y\times X\times Y && \\ & X\times X\times Y \arrow[u,hook,swap,"j"] \arrow[ld,swap,"q"] \arrow[rd] \\ X\times X \arrow[rd] & \commsquare[45] & X\times Y \arrow[ld] \ar[rd] \\ & X && Y \end{tikzcd}$$

\end{document}