diagram composing function

Question

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

But the most I got is

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-> to node {$\tau_{c}^{-1}$}(2);
    \draw-> to node {$l^{-1}$}(3);
    \draw-> to node {$\tau_{-1}$}(4);
    \draw-> to node {$f^{}$}(5);
    \draw- to node {}(11);
    \draw-> to node {}(5);
    \draw- to node {$f = \tau_{c}^{-1} \circ l^{-1} \circ \tau_{-1} \circ f^{*}$}(15);
\end{tikzpicture}
\end{document}

Thanks

Answer 1

You were almost at it.

First version with too much vertical space:

Second one with reduced vertical space:

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-> to node {$\tau_{c}^{-1}$}(2);
    \draw-> to node {$l^{-1}$}(3);
    \draw-> to node {$\tau_{-1}$}(4);
    \draw-> 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}

Answer 2

Alternative possibility where are employ tikz libraries arrows.meta (for arrows), chains (for make chain of nodes), positioning (for distances between nodes) and quotes )for edge lables):

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta, 
                chains,
                positioning,
                quotes}
\begin{document}
    \begin{tikzpicture}[auto,
node distance = 8mm and 12mm, 
  start chain = A going right,
every edge/.style = {draw, -Straight Barb}
                        ]
\foreach \i in {C, C_0, P_0, P, f^{}(P)}
    \node[on chain=A] {$\i$};
\coordinate[below=of A-5]  (A-6);
\draw   (A-1) edge["$\tau_{c}^{-1}$"]   (A-2) 
        (A-2) edge["$l^{-1}$"]          (A-3) 
        (A-3) edge["$\tau_{-1}$"]       (A-4) 
        (A-4) edge["$f^{}$"]           (A-5)
        (A-1) |- (A-6)  node[pos=0.75, below] 
                        {$f = \tau_{c}^{-1} \circ l^{-1} \circ \tau_{-1} \circ f^{*}$} 
        (A-6) edge (A-5);
    \end{tikzpicture}
\end{document}

Answer 3

The code is quite short with a normal displaymath environment, using \xrightarrow and pstricks(more precisely with pst-node):

\documentclass{article}
\usepackage{amsmath}
\usepackage{pst-node}
\begin{document}
[ \rnode{C}{C} \xrightarrow[\rule{3em}{0pt}]{\tau_c^{-1}} C_0 \xrightarrow[\rule{3em}{0pt}]{l^{-1}}P_0 \xrightarrow[\rule{3em}{0pt}]{\tau_{-1}}P 0\xrightarrow[\rule{3em}{0pt}]{f}\rnode{F}{f^(P)}
  \psset{linewidth=0.5pt, angle = -90, arm=0.8cm, arrowinset=0.15, nodesepA=3pt, linejoin=1}
  \ncbar{->}{C}{F} \nbput{f^*\circ \tau_{-1}\circ l^{-1}\circ \tau_c^{-1}}
  ]%
\end{document}