mjhjmtu

I want to use tikzmark in tikzcd.
However, even the following sample returns errors

documentclassarticle
usepackageamsmath
usepackagetikz-cd 
usetikzlibrarytikzmark

begindocument

begintikzcd
tikzmarku1A & B\
C & tikzmarku2D
endtikzcd

begintikzpicture[remember picture,overlay]
draw[->] (u1) edge (u2);
endtikzpicture
enddocument

The error report is

Package tikz Error: Cannot parse this coordinate. tikzmarku1A

Package tikz Error: Cannot parse this coordinate. C & tikzmarku2D

Package pgf Error: No shape named u1 is known. draw[->] (u1)

Package pgf Error: No shape named u1 is known. draw[->] (u1) edge
(u2)

Package pgf Error: No shape named u2 is known. draw[->] (u1) edge
(u2)

Package pgf Error: No shape named u2 is known. draw[->] (u1) edge
(u2)

Package pgf Error: No shape named u1 is known. draw[->] (u1) edge
(u2)

Label(s) may have changed. Rerun to get cross-references right.

How to fix this kind of errors?

Update:
I have tried this

documentclassarticle
usepackageamsmath
usepackagetikz-cd 
usetikzlibrarytikzmark

begindocument

begintikzcd[remember picture]
aaaaaasubnodeu1A & B\
C & subnodeu2Dddddddd
endtikzcd

begintikzpicture[remember picture,overlay]
draw[->] (u1) edge (u2);
endtikzpicture
enddocument

and this

documentclassarticle
usepackageamsmath
usepackagetikz-cd 
usetikzlibrarytikzmark

begindocument

begintikzcd[remember picture]
aaaaaatikzmarknodeu1A & B\
C & tikzmarknodeu2Dddddddd
endtikzcd

begintikzpicture[remember picture,overlay]
draw[->] (u1) edge (u2);
endtikzpicture
enddocument

But none of them put the arrow in correct location.

Update:
The tikzmarknode command does work correctly if I use LuaLaTex instead of XeLaTeX.

(Note: I agree with marmot that for this particular case I wouldn't use tikzmark, but I get that it is a simple example and it reveals a bug in tikzmark.)

It would appear that when I modified tikzmark to make it possible to use inside tikzpictures, then I inadvertently broke the ability to use it inside a node (though I'm not sure if that was ever a recommended use!). A node is inside a tikzpicture but should be treated as if it weren't. My current test for whether the tikzmark is inside a tikzpicture doesn't pick up on this. Here's the corrected test:

documentclassarticle
%urlhttps://tex.stackexchange.com/q/458387/86
usepackageamsmath
usepackagetikz-cd 
usetikzlibrarytikzmark

makeatletter

deftikzmark%
 ifxpgfpictureid@undefined
 lettikzmark@next=tikzmark@outside
 else
 relax
 ifxscopetikz@origscoperelax
 lettikzmark@next=tikzmark@outside
 else
 lettikzmark@next=tikzmark@inside
 fi
 fi
 tikzmark@next%

makeatother

begindocument

begintikzcd
tikzmarku1A & B\
C & tikzmarku2D
endtikzcd

begintikzpicture[remember picture,overlay]
draw[->] (pic cs:u1) -- (pic cs:u2);
endtikzpicture
enddocument

I've uploaded this to github and will send it off to CTAN once the amazing testing team of marmot has answered a few questions to check that I haven't broken something else!

REVISED ANSWER: I could get tikzmark to collaborate with tikz-cd by switching to tikzmarknode. More precisely, if I rewrite cfr's nice answer to become

documentclassarticle
usepackagetikz-cd,amssymb
usetikzlibrarytikzmark,quotes
begindocument
begintikzcd
 x in tikzmarknodeu1U subseteq M arrowrf & f(x) in V subseteq 
 tikzmarknodenN arrowdpsi \
 phi(x) in tikzmarknodeu2phi(U) subseteq mathbbR^m arrowrpsi f phi^-1 & psi(f(x)) in psi(V) subseteq mathbbR^n
endtikzcd
begintikzpicture[overlay, remember picture]
 draw [->] (u1) edge ["$phi$", right] (u2.north -| u1);
endtikzpicture
enddocument

I get

which might be what you are after.

Let me also mention that for your above example you may not use tikzmark since you have the objects already in TikZ nodes.

documentclassarticle
usepackageamsmath
usepackagetikz-cd 

begindocument

begintikzcd[remember picture]
A & B\
C & D
endtikzcd

begintikzpicture[remember picture,overlay]
draw[->] (tikzcdmatrixname-1-1) edge (tikzcdmatrixname-2-2);
endtikzpicture
enddocument