Trying to draw this commutative diagram: Tex

I have been able to draw the right part:

& (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

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


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

\[\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

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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