How to write composition of epimorphisms

Question

I am trying to write the following composition of epimorphisms, but I don't like the form.

The code is:

\documentclass[12pt,a4paper]{book}
\usepackage[utf8]{inputenc}
\usepackage[english,greek]{babel}
\usepackage{amsmath,amsfonts,amssymb}

\newcommand{\rightarrowdbl}{\rightarrow\mathrel{\mkern-14mu}\rightarrow}
\newcommand{\xrightarrowdbl}[2][]{%
  \xrightarrow[#1]{#2}\mathrel{\mkern-14mu}\rightarrow}

\begin{document}
\begin{align*}
\phi \circ \pi: \mathbb{Z}[X] & \xrightarrowdbl{\ \pi \ } \mathbb{Z}_2[X] && \xrightarrowdbl{\ \phi \ }  \mathbb{Z}_2,\\
f(X) & \longmapsto \pi (f(X)):=\overline{f(X)} && \longmapsto  \phi(\overline{f(X)}):=\overline{f(0)}
\end{align*}
\end{document}

Is there a better way to write it?

Thank you!

Answer 1

I will chose for to draw your composition of epimorphisms, the classic package to create commutative diagrams: tikz-cd.

The use of tikz-cd it makes the reading of switching diagrams better as the picture shown below.

\documentclass[a4paper,12pt]{article}
\usepackage{tikz-cd,mathtools,amssymb}

\begin{document}
\begin{tikzcd}[row sep=1cm, column sep=1cm]
{\phi \circ \pi: \mathbb{Z}[X]} \arrow[r, "\pi", two heads] & {\mathbb{Z}_2[X]} \arrow[r, "\phi", two heads] & \mathbb{Z}_2 
\end{tikzcd}

\begin{tikzcd}[row sep=1cm, column sep=.5cm]
f(X)   \arrow[r, maps to] & \pi (f(X))\mathrel{\mathop:}=\overline{f(X)} \arrow[r, maps to] & \phi(\overline{f(X)})\mathrel{\mathop:}=\overline{f(0)}
\end{tikzcd}
\end{document}