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How to group the blue objects $C_{n+1},C_n,D_{n+1},D_n$ in my codes in a parallelogram just like the following picture using the command '\save' ? Some examples

\documentclass{article}
\usepackage[all]{xy}
\begin{document}
$$\xymatrix{ 
    C:\ar[dd]_{\varphi}^{\psi} & \cdots\ar[r] & C_{n+1}\ar[rr]^{\partial_{n+1}^C}\ar[dd] && C_n\ar[rr]^{\partial_n^C}\ar[dd]_{\varphi_n}^{\psi_n}\ar[ddll]_{T_n} && C_{n-1}\ar[ddll]_{T_{n-1}}\ar[r]\ar[dd] & \cdots\\\\
    D: & \cdots\ar[r] & D_{n+1}\ar[rr]^{\partial_{n+1}^D} && D_n\ar[rr]^{\partial_n^D} && D_{n-1}\ar[r] & \cdots,}$$
\end{document}

What I need is grouping the following blue objects: What I need

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Not exactly the same diagram, and not exactly a parallelogram, but here's something similar using the tikz-cd package:

\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-cd}
\usetikzlibrary{tikzmark}
\begin{document}
\[
\begin{tikzcd}
  C:\ar[dd,"\varphi"',"\psi"] & \cdots\ar[r] &
  C_{n+1}\ar[rr,"\partial_{n+1}^C"]\ar[dd] &&
  C_n\ar[rr,"\partial_n^C"]\ar[dd,"\varphi_n"',"\psi_n"]\ar[ddll,"T_n"'] &&
  C_{n-1}\ar[ddll,"T_{n-1}"']\ar[r]\ar[dd] & \cdots\\\\
  D: & \cdots\ar[r] & D_{n+1}\ar[rr,"\partial_{n+1}^D"] &&
  D_n\ar[rr,"\partial_n^D",near end] && D_{n-1}\ar[r] & \cdots
  %
  \ar[from=1-5,to=1-7,color=blue,dash,start
  anchor={[xshift=-1em,yshift=3ex]center},end
  anchor={[xshift=3.8em,yshift=3ex]center}]
  \ar[from=3-3,to=3-5,color=blue,dash,start
  anchor={[xshift=-4em,yshift=-2ex]center},end
  anchor={[xshift=1em,yshift=-2ex]center}]
  \ar[from=3-3,to=1-5,color=blue,dash,start
  anchor={[xshift=-4em,yshift=-2ex]center},end
  anchor={[xshift=-1em,yshift=3ex]center}]
  \ar[from=1-7,to=3-5,color=blue,dash,start
  anchor={[xshift=3.8em,yshift=3ex]center},end
  anchor={[xshift=1em,yshift=-2ex]center}]
\end{tikzcd}
\]
\end{document}

The problem is that tikz-cd doesn't allow one to use usual tikz commands inside the diagram, so I've just modeled the parallelogram by four edges (which are drawn as arrows without tips). The result is:

enter image description here

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