How to draw composition of function?

Maybe with tikz-cd. I am not convinced that these ellipses help here, but it is your call.

\documentclass{article}
\usepackage{amssymb}
\usepackage{tikz-cd}
\usetikzlibrary{shapes.geometric}
\begin{document}
\[\tikzcdset{del/.style={end anchor=90+#1,start anchor=90-#1},
del/.default=2}
\begin{tikzcd}[cells={nodes={ellipse,minimum height=12em,draw}},
    bend angle=50,every arrow/.append style={-stealth}]
 M_n\times M_n \arrow[r,bend left,"R_i",del=5]
 \arrow[rrrr,bend right=30,"\succ",end anchor=-95,start anchor=-85]
 &  \mathrm{Row}_i\times \mathrm{Row}_i \arrow[r,bend left,"\omega_{ij}",del]
 & M_n \arrow[r,bend left,"\omega_{ij}'",del]
 & M_{1\times n} \arrow[r,bend left,"\omega^*",del]
 & \mathbb{R}\\
\end{tikzcd}
\]
\end{document}

As I said, ellipses may not be optimal here. So here are shaded circles of the appropriate size.

\documentclass{article}
\usepackage{amssymb}
\usepackage{tikz-cd}
\begin{document}
\[
\begin{tikzcd}[cells={nodes={circle,align=center,inner sep=1pt,
    text width={width("$\mathrm{Row}_i\times\mathrm{Row}_i$")},draw,
    path picture={\path[ball color=gray!60,fill opacity=0.4]
     (path picture bounding box.center) 
     circle[radius={width("$\mathrm{Row}_i\times\mathrm{Row}_i$")*1pt}];
    }}},
    bend angle=50,every arrow/.append style={-stealth,bend left},
    every label/.append style={font=\normalsize},
    sep=small]
 M_n\times M_n \arrow[r,"R_i"]
 \arrow[rrrr,bend right=25,"\succ",start anchor=-60,end anchor=-120]
 &  \mathrm{Row}_i\times \mathrm{Row}_i \arrow[r,"\omega_{ij}"]
 & M_n \arrow[r,"\omega_{ij}'"]
 & M_{1\times n} \arrow[r,"\omega^*"l]
 & \mathbb{R}\\
\end{tikzcd}
\]
\end{document}

To my taste diagram is nicer with equal sized circles instead of ellipses. Using tikz-cd (similarly as proposed in Schrödinger's cat answer), the MWE is:

\documentclass{article}
\usepackage{amssymb}
\usepackage{tikz-cd}

\begin{document}
    \[
\begin{tikzcd}[sep=small,
               cells={nodes={circle, draw, minimum size=7em, inner sep=1pt}}, 
               every arrow/.append style = {-stealth, bend left=60}
               every label/.append style = {font=\normalsize}
               ]
M_n\times M_n \ar[r,"R_i"]
              \ar[rrrr,bend right=40, "\succ" ']
    &   \mathrm{Row}_i\times\mathrm{Row}_i \ar[r,"\omega_{ij}"]
        &   M_n \ar[r,"\omega_{ij}'"]
            &   M_{1\times n} \ar[r,"\omega^*"]
                &   \mathbb{R}  \\  
\end{tikzcd}
    \]
\end{document}

I don't know the package tikz-cd. Let's see what we can do with TikZ.

Step 1 :

The next minimum code is enough to draw what you want

\documentclass[border=.25cm]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric }
\begin{document}
\begin{tikzpicture}
\foreach \i in {1,...,5}{%
\node[draw,ellipse,minimum height=3cm, minimum width=2cm] (E\i) at (2.5*\i,0)  {};}
\draw[->] (E1.north) to[bend left=45]  node[auto] {$R$} (E2.north);
\draw[->] (E1.south) to[bend right=45] node[auto] {$>$} (E5.south);
\end{tikzpicture}

\end{document}

Step 2 :

\documentclass[border=.25cm]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric }
\usepackage{amssymb}
\begin{document}
\begin{tikzpicture}
\foreach \i/\content in {1/$M_n\times M_n$,2/$Row_i\times Row_i$,3/$M_n$,4/$M_{1\times n}$,5/$\mathbb{R}$}{%
\node[draw,ellipse,minimum height=3cm, minimum width=1cm] (E\i) at (4*\i,0)  {\content };}
\draw[->] (E1.north) to[bend left=45]  node[auto] {$R$} (E2.north);
\draw[->] (E1.south) to[bend right=45] node[auto] {$>$} (E5.south);
\end{tikzpicture}

\end{document}

Step final :

\documentclass[border=.25cm]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric }
\usepackage{amssymb}
\begin{document}
\begin{tikzpicture}
\foreach \i/\content in {1/$M_n\times M_n$,2/$Row_i\times Row_i$,3/$M_n$,4/$M_{1\times n}$,5/$\mathbb{R}$}{%
\node[draw,ellipse,minimum height=3cm, minimum width=.8cm] (E\i) at (3.2*\i,0)  {\content };}
\foreach \i/\lb [count=\j from 2] in {1/$R_i$,2/$\omega_{ij}$,3/$\omega_{ij}'$,4/$\omega^*$}{%
\draw[->] (E\i.north) to[bend left=45]  node[auto] {\lb} (E\j.north);}
 \draw[->] (E1.south) to[bend right=45] node[auto] {$>$} (E5.south);
\end{tikzpicture}

\end{document}