Mfcttrf

I have this diagram

 begintikzcd
 & N arrow[twoheadrightarrow]dpi\
 R^bigoplusmathcalI arrowrgamma arrowurbeta & M 
 endtikzcd

Now I want to define $gamma$ and $beta$ in the diagram. I mean, I want to replace $gamma$ and $beta$ in the diagram by

beginarrayrcl
 gamma colon R^bigoplusmathcalI & longrightarrow & M\
 e_i & longmapsto & gamma(e_i)
endarray

and

beginarrayrcl
beta colon R^bigoplusmathcalI & longrightarrow & N\
 e_i & longmapsto & g_i
endarray

Finally, I also want to add this

beginarrayrcl
pi^-1 colon M & longrightarrow & N\
 gamma(e_i) & longmapsto & g_i
endarray

(This means preimage, obv $pi$ may not be bijective)Any ideas?(when replacing the arrows, I want to keep the arrow's names
above the arrows). I want this diagram to check that $R^bigoplusmathcalI$ (free module over mathcalI) is a projective module.

For fun, a solution with the psmatrix environment, from pstricks:

documentclassarticle
usepackageamsmath
 usepackagepst-node

begindocument
%
 [
 everypsboxscriptstyle
 pssetarrows=->, arrowinset=0.15, linewidth=0.4pt, nodesep=3pt, labelsep=2pt, rowsep=1.2cm, colsep=1.5cm, shortput=nab
 beginpsmatrix
%%%nodes
 & [name=N] N \%
R^ oplus mathcalIpnodeR & [name=M] M
 endpsmatrix
%%% arrows
 ncline2,11,2naput[npos=0.45,labelsep =0pt]beta 
 ncline2,12,2^gamma 
 ncline1,22,2_pi 
%%%%
 psset->,offset =2.5ex
 ncline2,11,2
uput4ex[125]45(R)e_i
uput2ex[140]45(N)g_i
%%%
 ncline[offset=-2ex]2,12,2
uput1ex[-110]0(R)e_i
uput1ex[-60]0(M)gamma (e_i)
%%%
 ncline[linestyle=dashed, arrows=<-|]1,22,2naput[npos=0.43]pi^-1
uput2ex[r]0(N)g_i
 ]

enddocument 

As long as you add an ampersand replacement you can use these arrays.

documentclassarticle
usepackageamsmath
usepackagetikz-cd
begindocument

begintikzcd[column sep=8em,ampersand replacement=&]
 & N arrow[twoheadrightarrow,d,"pi"]\
 R^bigoplusmathcalI arrow[r,"beginarrayrcl
 gamma colon R^bigoplusmathcalI & longrightarrow & M\
 e_i & longmapsto & gamma(e_i)
endarray" swap] 
arrow[ur,"beginarrayrcl
beta colon R^bigoplusmathcalI & longrightarrow & N\
 e_i & longmapsto & g_i
endarray" sloped] & M arrow[u,twoheadrightarrow,bend right=80,"beginarrayrcl
pi^-1 colon M & longrightarrow & N\
 gamma(e_i) & longmapsto & g_i
endarray" swap]
 endtikzcd
enddocument

Define a macro for the complicated thing:

documentclassarticle
usepackageamsmath
usepackagetikz-cd

newcommandfuncdef[5]%
 mboxscriptsize$
 beginaligned #1colon #2 & longrightarrow #3 \[-1ex] #4 &longmapsto #5 endaligned
 $%



begindocument

begintikzcd[column sep=8em]
& N arrow[twoheadrightarrow,d,"pi"]\
R^bigoplusmathcalI
 arrow[r,"funcdefgammaR^bigoplusmathcalIMe_igamma(e_i)"']
 arrow[ur,"funcdefbetaR^bigoplusmathcalINe_ig_i" sloped] &
M arrow[u,twoheadrightarrow,bend right=80,"funcdefpi^-1MNgamma(e_i)g_i"']
endtikzcd

enddocument