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Trying to draw this commutative diagram: Tex

I have been able to draw the right part:

\begin{tikzcd}
& (U_i\cap U_j)\times\mathbb{C}^n \arrow[dd,"id\times g_{ij}=\varphi_j\circ\varphi_i^{-1}"] \\
\pi^{-1}(U_i\cap U_j) \arrow[ur,"\varphi_i"] \arrow[dr,"\varphi_j"] & \\
& (U_i\cap U_j)\times\mathbb{C}^n
\end{tikzcd}

Can someone please help me to draw the left part of the diagram and attach it accordingly? Thanks in advance.

2

It is more or less copy and paste of the right part.

\documentclass{article}  
\usepackage[margin=1in]{geometry}
\usepackage{amssymb}
\usepackage{tikz-cd}
\begin{document}
\[\begin{tikzcd}[column sep=1em]
(U_i\cap U_j)\times\mathbb{C}^n \arrow[dd,"\psi_j\circ\psi_i^{-1} id\times g_{ij}" '] 
 & & & & 
(U_i\cap U_j)\times\mathbb{C}^n \arrow[dd,"id\times g_{ij}=\varphi_j\circ\varphi_i^{-1}"] \\
%
&  \pi^{-1}(U_i\cap U_j) \arrow[ul,"\psi_i"'] \arrow[dl,"\psi_j"] 
& &\pi^{-1}(U_i\cap U_j) \arrow[ll,"L"'] \arrow[ur,"\varphi_i"] \arrow[dr,"\varphi_j"] & \\
(U_i\cap U_j)\times\mathbb{C}^n & & & & (U_i\cap U_j)\times\mathbb{C}^n
\end{tikzcd}\]
\end{document}

enter image description here

Please note that, since you only post a code fragment, I had to make an assumption about your page settings. If your page layout is much narrower than what I assumed, you will have to tweak the diagram to fit on the page.

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