Ttdfjt

I am trying do make this looking better:

namely: curly brackets enclosing everything, tikzcd diagram vertically centered compared to the two other lines. Any suggestion?

documentclass[letter, 11pt]article
usepackagemultirow
usepackageamsmath
usepackagetikz-cd
tikzset
 commutative diagrams/.cd,
 arrow style=tikz,
 diagrams=>=Computer Modern Rightarrow[length=5pt,width=5pt],


begindocument
 beginalign*
 M := 
 leftbeginarrayc 
 f:A to B & multirow2*
 begintikzcd[ampersand replacement=&,column sep=1em]
 X times Y ar[r, "m"] ar[d, "r"']& Z times W ar[r, "n"] & V\
 X times Y ar[r, "m"] & Z times ar[r, "n"]W & V ar[u, "h"'] 
 endtikzcd 
 \
 g: A times X to Y & \
 endarrayright
 endalign*
enddocument

an alternative, simple solution:

documentclass[letter, 11pt]article
usepackageamsmath
usepackagetikz-cd
tikzset
 commutative diagrams/.cd,
 arrow style=tikz,
 diagrams=>=Computer Modern Rightarrow[length=5pt,width=5pt],


begindocument
 [
 M :=
 leftbeginarrayc
 begingathered
 f:A to B \
 g: A times X to Y
 endgathered & begintikzcd[ampersand replacement=&]
 X times Y ar[r, "m"] ar[d, "r"']& Z times W ar[r, "n"] & V\
 X times Y ar[r, "m"] & Z times ar[r, "n"]W & V ar[u, "h"']
 endtikzcd
 endarrayright
 ]

addendum: some off-topic remarks:

• 
  

  for determining arrows style you can instead of tikzset use (shorter) tikzcdset, for example:

  
  
  

  tikzcdsetarrow style=tikz,
   diagrams=>=Straight Barb % I liked such arrows :-)
   
  

  
  



• in your case you not need ampersand replacement=&



• 
  

  arrows is better -- due to consistency of code -- to write after node content, i.e.: instead Z times ar[r, "n"]W is better Ztimes W ar[r, "n"], regardless that resulting diagram is the same

  
  
  

  documentclass[margin=3mm, varwidth]standalone
  usepackageamsmath
  usepackagetikz-cd
  tikzcdsetarrow style=tikz,
   diagrams=>=Straight Barb
   
  
  begindocument
   [
  M := leftbeginarrayc
   begingathered
   f:A to B \
   g: A times X to Y
   endgathered & begintikzcd%[sep=large] % i like bigger diagram :-)
   Xtimes Y ar[r, "m"] ar[d, "r"'] & Ztimes W ar[r, "n"] & V\
   Xtimes Y ar[r, "m"] & Ztimes W ar[r, "n"] & V ar[u, "h"']
   endtikzcd
   endarrayright
   ]
  enddocument
  

  
  

gives:

I think you should not use left and right.

Also, letter is not defined yet. Use letterpaper.

documentclass[letterpaper,11pt]article
usepackagemathtools
usepackagetikz-cd
tikzset
 commutative diagrams/.cd,
 arrow style=tikz,
 diagrams=>=Computer Modern Rightarrow[length=5pt,width=5pt],

makeatletter
newcommandvastbBigg@4
makeatother
begindocument
[
Mcoloneqqvastbegintikzcd[ampersand replacement=&,column sep=1em]
X times Y ar[r, "m"] ar[d, "r"']& Z times W ar[r, "n"] & V\
X times Y ar[r, "m"] & Z times ar[r, "n"]W & V ar[u, "h"'] 
endtikzcd
vast
]
enddocument

The arrow tips are not consistent at all... However, as that is your intention, I keep it.

documentclass[letter, 11pt]article
usepackageamsmath
usepackagetikz-cd
tikzset
 commutative diagrams/.cd,
 arrow style=tikz,
 diagrams=>=Computer Modern Rightarrow[length=5pt,width=5pt],


begindocument
 beginalign*
 M := left
 begintikzcd[ampersand replacement=&,column sep=1em]
 X times Y ar[r, "m"] ar[d, "r"']& Z times W ar[r, "n"] & V\
 X times Y ar[r, "m"] & Z times ar[r, "n"]W & V ar[u, "h"'] 
 endtikzcd
 right
 endalign*
enddocument