tikzcd diagram within an array

an alternative, simple solution:

\documentclass[letter, 11pt]{article}
\usepackage{amsmath}
\usepackage{tikz-cd}
\tikzset{
    commutative diagrams/.cd,
    arrow style=tikz,
    diagrams={>={Computer Modern Rightarrow[length=5pt,width=5pt]}},
}

\begin{document}
    \[
        M :=
        \left\{\begin{array}{c|c}
        \begin{gathered}
        f:A \to B            \\
        g: A \times X \to Y
        \end{gathered} &  \begin{tikzcd}[ampersand replacement=\&]
                                X \times Y \ar[r, "m"] \ar[d, "r"']\& Z \times W \ar[r, "n"] \& V\\
                                X \times Y \ar[r, "m"] \& Z \times \ar[r, "n"]W \& V    \ar[u, "h"']
                                            \end{tikzcd}
        \end{array}\right\}
    \]
    \end{document}

addendum: some off-topic remarks:

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

  \tikzcdset{arrow 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 Z\times W \ar[r, "n"], regardless that resulting diagram is the same

  \documentclass[margin=3mm, varwidth]{standalone}
  \usepackage{amsmath}
  \usepackage{tikz-cd}
  \tikzcdset{arrow style=tikz,
             diagrams={>=Straight Barb}
             }
  
  \begin{document}
      \[
  M := \left\{\begin{array}{c|c}
      \begin{gathered}
      f:A \to B            \\
      g: A \times X \to Y
      \end{gathered} & \begin{tikzcd}%[sep=large]  % i like bigger diagram :-)
                          X\times Y \ar[r, "m"] \ar[d, "r"'] & Z\times W \ar[r, "n"] & V\\
                          X\times Y \ar[r, "m"]              & Z\times W \ar[r, "n"] & V \ar[u, "h"']
                       \end{tikzcd}
      \end{array}\right\}
      \]
  \end{document}
  

gives:

I think you should not use \left and \right.

Also, letter is not defined yet. Use letterpaper.

\documentclass[letterpaper,11pt]{article}
\usepackage{mathtools}
\usepackage{tikz-cd}
\tikzset{
    commutative diagrams/.cd,
    arrow style=tikz,
    diagrams={>={Computer Modern Rightarrow[length=5pt,width=5pt]}},
}
\makeatletter
\newcommand{\vast}{\bBigg@{4}}
\makeatother
\begin{document}
\[
M\coloneqq\vast\{
\begin{array}{c}
    f:A\to B\\
    g:A\times X\to Y
\end{array}\vast|\begin{tikzcd}[ampersand replacement=\&,column sep=1em]
X \times Y \ar[r, "m"] \ar[d, "r"']\& Z \times W \ar[r, "n"] \& V\\
X \times Y \ar[r, "m"] \& Z \times \ar[r, "n"]W \& V    \ar[u, "h"']    
\end{tikzcd}
\vast\}
\]
\end{document}

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

\documentclass[letter, 11pt]{article}
\usepackage{amsmath}
\usepackage{tikz-cd}
\tikzset{
    commutative diagrams/.cd,
    arrow style=tikz,
    diagrams={>={Computer Modern Rightarrow[length=5pt,width=5pt]}},
}

\begin{document}
    \begin{align*}
     M :=   \left\{
                \begin{array}{ll}
                    f:  & A \to B           \\ 
                    g:  & A \times X \to Y  \\
                \end{array}
                \middle|
                \begin{tikzcd}[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"']        
                \end{tikzcd}
            \right\}
    \end{align*}
\end{document}