How to draw a commutative diagram like this one?

Question

I need a diagram like this one:

This is what I've got so far:

    \documentclass{article}
    \usepackage{amsmath}
    \usepackage{tikz}
    
    \begin{document}
    
    \begin{tikzpicture}[node distance=2cm, auto]
    \node (Lambda1) {$\Lambda$};
    \node(Sigma1) [right of=Lambda1] {$\Sigma_2$};
    \node (Lambda2) [below of=Lambda1] {$\Lambda$};
    \node (Sigma2) [below of=Sigma1] {$\Sigma_2$};
    \draw[->](Lambda1) to node {$S$}(Sigma1);
    \draw[->](Lambda2) to node [left] {$f_\mu$}(Lambda1);
    \draw[->](Sigma2) to node [right] {$\tau$}(Sigma1);
    \draw[->](Lambda2) to node [below] {$S$}(Sigma2);
    \end{tikzpicture}
    
    \end{document}

Result:

I don't know how to insert those curved arrows with the compositions:

$ S \circ f_\mu$ and $\tau \circ S$.

Answer 1

With a bit enlarged distance between nodes and the same text for better observing the behaviour:

\usepackage{tikz}

\begin{document}

\begin{tikzpicture}[node distance=3cm, auto]
\node (Lambda1) {$\Lambda$};
\node(Sigma1) [right of=Lambda1] {$\Sigma_2$};
\node (Lambda2) [below of=Lambda1] {$\Lambda$};
\node (Sigma2) [below of=Sigma1] {$\Sigma_2$};
\draw[->](Lambda1) to node {$S$}(Sigma1);
\draw[->](Lambda2) to node [left] {$f_\mu$}(Lambda1);
\draw[->](Sigma2) to node [right] {$\tau$}(Sigma1);
\draw[->](Lambda2) to node [below] {$S$}(Sigma2);

\draw[->](Lambda2) to  [bend left=30] node [right,xshift=2pt] {$S\circ f_\mu$} (Sigma1);
\draw[->](Lambda2) to  [bend right=30] node [ left, xshift=-2pt] {$S\circ f_\mu$} (Sigma1);


\end{tikzpicture}

Answer 2

Direct method with tikz-cd.

\documentclass{article}

\usepackage{tikz-cd}

\begin{document}

\[
  \begin{tikzcd}
    \Lambda & {\Sigma_2} \\
    \Lambda & {\Sigma_2}
    \arrow["S", from=1-1, to=1-2]
    \arrow["S"', from=2-1, to=2-2]
    \arrow["{f_\mu}", from=2-1, to=1-1]
    \arrow["\tau"', from=2-2, to=1-2]
    \arrow["{\scriptstyle S \circ f_\mu}"{marking}, bend left = 30pt, from=2-1, to=1-2]
    \arrow["{\scriptstyle \tau \circ S}"{marking}, bend right = 30pt, from=2-1, to=1-2]
  \end{tikzcd}
\]

\end{document}

Better visual effect if you have quiver.sty:

\documentclass{article}

\usepackage{quiver}

\begin{document}

\[
  \begin{tikzcd}
    \Lambda & {\Sigma_2} \\
    \Lambda & {\Sigma_2}
    \arrow["S", from=1-1, to=1-2]
    \arrow["S"', from=2-1, to=2-2]
    \arrow["{f_\mu}", from=2-1, to=1-1]
    \arrow["\tau"', from=2-2, to=1-2]
    \arrow["{\scriptstyle S \circ f_\mu}"{marking}, shift right=1, curve={height=-12pt}, from=2-1, to=1-2]
    \arrow["{\scriptstyle \tau \circ S}"{marking}, shift left=1, curve={height=12pt}, from=2-1, to=1-2]
  \end{tikzcd}
\]

\end{document}

If you prefer the labels off the arrow, you may shift the labels along the y-axis (notice the yshift):

\begin{tikzcd}
    \Lambda & {\Sigma_2} \\
    \Lambda & {\Sigma_2}
    \arrow["S", from=1-1, to=1-2]
    \arrow["S"', from=2-1, to=2-2]
    \arrow["{f_\mu}", from=2-1, to=1-1]
    \arrow["\tau"', from=2-2, to=1-2]
    \arrow["{\scriptstyle S \circ f_\mu}"{marking, yshift=-5pt}, shift right=1, curve={height=-12pt}, from=2-1, to=1-2]
    \arrow["{\scriptstyle \tau \circ S}"{marking, yshift=5pt}, shift left=1, curve={height=12pt}, from=2-1, to=1-2]
\end{tikzcd}