Arrows between factors of a product in \tikzcd

Question

I would like to incorporate this diagram into my essay

I am working with the \tikzcd package, and I have no clue how to make the diagram in that environment. The only problem is with the arrows that represent maps between the factors of the corresponding products. How can this be made?

Thanks in advance for your answers.

Answer 1

You can insert the \times symbols as arrows and act on the column separations.

\documentclass[a4paper,12pt]{article}
\usepackage{newtxtext,newtxmath}
\usepackage{mathtools}
\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{\sep}{\,|\,}

\begin{document}

\begin{tikzcd}[row sep=1ex]
H_n (M\sep L; R) \arrow[r,symbol=\times] \arrow[dd, "i_*"] &[-2em]
  H^k (M\sep L; R) \arrow[rd, "\frown"] &[2em]
\\
&& H_{n-k}(M;R) \\
H_n (M\sep K; R) \arrow[r,symbol=\times] &
  H^k (M\sep K; R) \arrow[ru,  "\frown"'] \arrow[uu, "i^*"']
\end{tikzcd}

\end{document}

Answer 2

I add my version using tikz-cd associated with the font clone of Times called newtxtext.

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


\begin{document}

\begin{tikzcd}[row sep=.2cm, column sep=.7cm]
H_n (M\mid L; R)\arrow[dd, "i_*"] & \mkern-40mu  \times \, H^k (M\mid L; R) \arrow[rrd, "\frown"] 
&  &  \\
 & &  & H_{n-k}(M;R) \\
H_n (M\mid K; R) & \arrow[uu, "i^*"'] \mkern-40mu\times \, H^k (M\mid K; R) \arrow[rru,  "\frown"'] &  &   
\end{tikzcd}

\end{document}   

Answer 3

\documentclass{article}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
  A \times B \arrow[d,xshift=-1em]\\
  C \times D \arrow[u,xshift=1em] 
\end{tikzcd}
\end{document}