Diagram composing function

diagramstikz-pgf

I'm trying to make a typical composing-function diagram and I've saw a few posts, but I cannot make it yet as I'd like. In this picture I show what I attempt to make
enter image description here

But the most I got is

enter image description here

As you can see, I used empty nodes for the broken line, but not exactly worthy. I leave my code here so you can help me with its edition. I must confess I'm basic user in Latex and 100% newbie in Tikz, so any guide you use is welcome.

\documentclass{article}

\usepackage{amsmath}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}[node distance=2cm, auto]
    \node(1) {$C$};
    \node(2) [right of=1] {$C_{0}$};
    \node(3) [right of=2] {$P_{0}$};
    \node(4) [right of=3] {$P$};
    \node(5) [right of=4] {$f^{*}(P)$};
    \node(11) [below of=1] {};
    \node(15) [below of=5] {};
    \draw[->](1) to node {$\tau_{c}^{-1}$}(2);
    \draw[->](2) to node {$l^{-1}$}(3);
    \draw[->](3) to node {$\tau_{-1}$}(4);
    \draw[->](4) to node {$f^{*}$}(5);
    \draw[-](1) to node {}(11);
    \draw[->](15) to node {}(5);
    \draw[-](11) to node {$f = \tau_{c}^{-1} \circ l^{-1} \circ \tau_{-1} \circ f^{*}$}(15);

\end{tikzpicture}

\end{document}

Thanks

Best Answer

You were almost at it.

First version with too much vertical space: pic

Second one with reduced vertical space: V2

You can adjust everything by paying attention to the code. The important part is --++(0,-1) here which draws a vertical line of 1cm length.

\documentclass{article}

\usepackage{amsmath}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}[node distance=2cm, auto]
    \node(1) {$C$};
    \node(2) [right of=1] {$C_{0}$};
    \node(3) [right of=2] {$P_{0}$};
    \node(4) [right of=3] {$P$};
    \node(5) [right of=4] {$f^{*}(P)$};
    \node(11) [below of=1] {};
    \node(15) [below of=5] {};
    \draw[->](1) to node {$\tau_{c}^{-1}$}(2);
    \draw[->](2) to node {$l^{-1}$}(3);
    \draw[->](3) to node {$\tau_{-1}$}(4);
    \draw[->](4) to node {$f^{*}$}(5);

    % \draw[->] (1) --++ (0,-2) -| node[pos=0.25] {$f = \tau_{c}^{-1} \circ l^{-1} \circ \tau_{-1} \circ f^{*}$} (5); %(old version)

    \draw[->] (1) --++ (0,-1) -| node[pos=0.25,below] {$f = \tau_{c}^{-1} \circ l^{-1} \circ \tau_{-1} \circ f^{*}$} (5);

\end{tikzpicture}

\end{document}
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