Achieve desired alignment of arrows in tikz-cd diagram

Question

I want to add to add the supersets of the sets U_\tau and V_\tau (respectively for \pi) to the following tikz-cd diagram

I managed to achieve the following result

Latex code:

\documentclass[openany]{book}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{tikz-cd}

\newcommand{\AAA}{\mathbb{A}}
\DeclareMathOperator{\Bl}{Bl}

\begin{document}

\begin{equation*}
    \begin{tikzcd}[/tikz/column 1/.append style={anchor=base east}, /tikz/column 2/.append style={anchor=base west}]
        S \supset U_\tau \arrow[r, "\varphi_0"] \arrow[d, "\tau", "\sim"'] & U_\pi \subset T \arrow[d, "\pi", "\sim"']\\
        \Bl_{(0,0)}(\AAA^2) \supset V_\tau \arrow[r, "\epsilon"] & V_\pi \subset \AAA^2
    \end{tikzcd}
\end{equation*}

\end{document}

I would like the arrows to be typeset as in the first diagram, i.e. going from U_\tau to V_\tau (respectively for \pi). I know i can move them manually, but i can't seem to get rid of the rotations.
Is there a way to tell tikz-cd that i want the arrows to start centered on U_\tau and end centered on V_\tau (respectively for \pi)?

Answer 1

Add “symbol” arrows, plus a trick to make the first column right aligned.

\documentclass{article}
\usepackage{mathtools,amssymb,calc}
\usepackage{tikz-cd}

% see https://tex.stackexchange.com/a/216042/4427
\tikzset{
  symbol/.style={
    draw=none,
    every to/.append style={
      edge node={node [sloped, allow upside down, auto=false]{$#1$}}}
  }
}

\newcommand{\AAA}{\mathbb{A}}
\DeclareMathOperator{\Bl}{Bl}

\begin{document}

\begin{equation*}
\begin{tikzcd}[
]
  \mathmakebox[\widthof{$\Bl_{(0,0)}(\AAA^2)$}][r]{S} \arrow[r,symbol=\supset] &[-1.5em]
  U_\tau \arrow[r, "\varphi_0"] \arrow[d, "\tau", "\sim"'] &
  U_\pi \arrow[r,symbol=\subset] \arrow[d, "\pi", "\sim"'] &[-1.5em]
  T 
\\
  \Bl_{(0,0)}(\AAA^2) \arrow[r,symbol=\supset] &
  V_\tau \arrow[r, "\epsilon"] &
  V_\pi \arrow[r,symbol=\subset] &
  \AAA^2
\end{tikzcd}
\end{equation*}

\end{document}