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}