Etydhv

I am having trouble with the label locations on my commutative diagram.

Here is my code:

documentclassarticle

usepackagetikz-cd

begindocument

begintikzcd[ampersand replacement = &]
 0 arrow[r, "Nlambda", bend left] & 1 arrow[r, "(N - 1)lambda", bend left] arrow[l, "lambda", bend left] & 2 arrow[r, "(N - 2)lambda", bend left] arrow[l, "2lambda", bend left] & cdots arrow[r, "2lambda", bend left] arrow[l, "3lambda", bend left] & N-1 arrow[r, "lambda", bend left] arrow[l, "(N-1)lambda", bend left] & N arrow[l, "Nlambda", bend left]
endtikzcd

begintikzcd[ampersand replacement = &]
 0 arrow[r, "Nlambda", bend left] & 1 arrow[r, "(N - 1)lambda", bend left] arrow[l, "lambda", bend left] & 2 arrow[r, "(N - 2)lambda", bend left] arrow[l, "2lambda", bend left] & 3 arrow[r, "2lambda", bend left] arrow[l, "3lambda", bend left] & 4 arrow[r, "lambda", bend left] arrow[l, "(N-1)lambda", bend left] & 5 arrow[l, "Nlambda", bend left]
endtikzcd

enddocument

which produces

I would like the arrow labels to appear "centered" on the arrows, as they are in the second diagram, but with the node labels from the first diagram. How can I fix this?

Note: I have already looked at the question here and know that I could probably manually correct the spacing on every single label, but that seems way more tedious than should be necessary given that I'm pretty sure the labels should be centered on the arrows by default.

What happens here? auto gets screwed up.

Why does it happen? See below.

Can one fix it? Yes, at least in an ad hoc way.

documentclassarticle
usepackagetikz-cd
tikzcdsetad hoc fix/.style=r'/.style=r,/tikz/above,l'/.style=l,/tikz/below,
every label/.append style=/tikz/auto=false

begindocument
begintikzcd[ampersand replacement=&,ad hoc fix]
 0 arrow[r', "Nlambda", bend left] & 
 1 arrow[r', "(N - 1)lambda", bend left] arrow[l', "lambda", bend left] & 
 2 arrow[r', "(N - 2)lambda", bend left] arrow[l', "2lambda", bend left] & 
 cdots arrow[r', "2lambda", bend left] arrow[l', "3lambda", bend left] & 
 N-1 arrow[r', "lambda", bend left] arrow[l', "(N-1)lambda", bend left] & 
 N arrow[l', "Nlambda", bend left]
endtikzcd

begintikzcd[ampersand replacement = &]
 0 arrow[r, "Nlambda", bend left] & 1 arrow[r, "(N - 1)lambda", bend left] arrow[l, "lambda", bend left] & 2 arrow[r, "(N - 2)lambda", bend left] arrow[l, "2lambda", bend left] & 3 arrow[r, "2lambda", bend left] arrow[l, "3lambda", bend left] & 4 arrow[r, "lambda", bend left] arrow[l, "(N-1)lambda", bend left] & 5 arrow[l, "Nlambda", bend left]
endtikzcd
enddocument

The reason why this happens is that the arrows have different vertical positions of start and end points. This can be seen from

documentclassarticle

usepackagetikz-cd

begindocument

begintikzcd[ampersand replacement = &,every label/.append style=sloped]
 0 arrow[r, "Nlambda", bend left] & 
 1 arrow[r, "(N - 1)lambda", bend left] arrow[l, "lambda", bend left] & 
 2 arrow[r, "(N - 2)lambda", bend left] arrow[l, "2lambda", bend left] & 
 cdots arrow[r, "2lambda", bend left] arrow[l, "3lambda", bend left] & 
 N-1 arrow[r,"lambda", bend left] arrow[l, "(N-1)lambda", bend left] & 
 N arrow[l, "Nlambda", bend left]
endtikzcd

enddocument

Misplacement of arrow labels you can be reduce by increasing the bend angles. With this you slightly change the position of the start and end of the arrows for longer nodes' names. For example:

documentclassarticle
usepackagetikz-cd

begindocument
 begintikzcd[sep=large,
 every arrow/.style=draw,->,bend left=45, % for shorter diagram code
 ]
0 ar[r, "Nlambda"]
 & 1 ar[r, "(N - 1)lambda"]
 ar[l, "lambda"]
 & 2 ar[r, "(N - 2)lambda"]
 ar[l, "2lambda"]
 & cdots ar[r, "2lambda"]
 ar[l, "3lambda"]
 & N-1 ar[r, "lambda"]
 ar[l, "(N-1)lambda"]
 & N ar[l, "Nlambda"]
 endtikzcd
enddocument

Looks like a good application for tikz's chain library :

documentclassarticle
usepackagetikz
usetikzlibraryautomata,chains,positioning
begindocument

begintikzpicture[
auto,
start chain = going right, 
state/.append style = 
 on chain,
 draw=none,
,
]
node (s0)[state] $0$;
node (s1)[state] $1$;
node (s2)[state] $2$;

draw[->] (s0) edge[bend left] node $Nlambda$ (s1)
 (s1) edge[bend left] node $lambda$ (s0)
 (s1) edge[bend left] node $(N-1)lambda$ (s2)
 (s2) edge[bend left] node $2lambda$ (s1)
 ;

endtikzpicture
enddocument