Control the alignment of each "cell" in amsmath's aligned environment

A tikz-cd attempt

\documentclass{article}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\sigma\colon\hspace*{-4.5em} & F(\alpha) \arrow{r}{\sim} \arrow[dash]{d}
& F'(\beta) \arrow[dash]{d} \\
\varphi\colon\hspace*{-4.5em} & F \arrow{r}{\sim}
& F'
\end{tikzcd}
\end{document} 

Surely the code can be improved, I'm not a tikz expert at all...

This is the result if you substitute dash with equal

You may be better off with tikz-cd or ..... But here, you can use a newcommand:

 \newcommand*{\mbc}[2]{\makebox[\widthof{$F(\alpha)$}][#1]{$#2$}}

which makes a box of width same as $F(\alpha)$ with the help of calc package.

\documentclass{article}
\usepackage{amsmath,calc}
\newcommand*{\mbc}[2]{\makebox[\widthof{$F(\alpha)$}][#1]{$#2$}}
\begin{document}
   \[
  \begin{aligned}
    \mbc{r}{\sigma} & \colon & \mbc{c}{F(\alpha)} & \overset{\sim}{\to} & F'(\beta)\\
           &        & \mbc{c}{|}     &                     & \mbc{c}{|}\\
    \phi    & \colon & \mbc{c}{F}     & \overset{\sim}{\to} & \mbc{c}{F'}
  \end{aligned}
  \]
\end{document}

Use c for center, l for left and r for right alignment.

For such a simple construction you can use the good old CD environment:

\documentclass{article}
\usepackage{amsmath,amscd}
\begin{document}
\[
\begin{CD}
\sigma\colon @. F(\alpha) @>\sim>> F'(\beta) \\
@. @| @| \\
\varphi\colon @. F @>\sim>> F'
\end{CD}
\]
\end{document}

However, it's better to switch to modern packages such as tikz-cd that are more flexible and powerful.