[Tex/LaTex] Can we rotate symbols in LaTeX? How should we make this diagram

Question

I would like to insert this diagram

Into my LaTeX document. However, I don't know if it is possible at all to rotate symbols, and how they could fit into a diagram (notice that the \cong symbol has been rotated 90 degrees counterclockwise). So, how could this diagram be made?

Thanks in advance for your interest.

Answer 1

Two solutions, one with “standard” tools, one with tikz-cd and code by LaRiFaRi.

\documentclass{article}
\usepackage{amsmath,amssymb} % for both solutions

\usepackage{graphicx} % for the first solution

\usepackage{tikz-cd} % for the second solution

% for the first solution
\newcommand{\overiso}[2]{%
  \overset{\substack{\textstyle #1\\[0.5ex]\rotcong\\[0.5ex]}}{#2}%
}
\newcommand{\rotcong}{\rotatebox[origin=c]{90}{$\cong$}}
% ===

% for the second solution, 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]{$#1$}}}
  }
}
% ===

% general commands
\newcommand{\numberset}[1]{\mathbb{#1}}
\newcommand{\Z}{\numberset{Z}}

\begin{document}

\[
\dots
\xrightarrow{\cong}
\overiso{\Z}{C_3(*)}
\xrightarrow{0}
\overiso{\Z}{C_2(*)}
\xrightarrow{\cong}
\overiso{\Z}{C_1(*)}
\xrightarrow{0}
\overiso{\Z}{C_0(*)}
\rightarrow
0
\]

\[
\begin{tikzcd}[row sep=1em]
& \Z & \Z & \Z & \Z \\
\dotsb \arrow[r,"\cong"] &
C_3(*) \arrow[r,"0"] \arrow[u,symbol=\cong] &
C_2(*) \arrow[r,"\cong"] \arrow[u,symbol=\cong] &
C_1(*) \arrow[r,"0"] \arrow[u,symbol=\cong] &
C_0(*) \arrow[r] \arrow[u,symbol=\cong] &
0
\end{tikzcd}
\]

\end{document}