# Vertical subset symbol in tikzcd

I am writing a simple commutative diagram and some objects of the second line are subsets of objects on the first line so instead of an upward arrow I would like to have a set inclusion symbol - which tilted 90 degrees to the left and effectively replacing the arrow. It seems easily achievable with a $\cong$ symbol but I cannot find an example with the subset or supset symbols.

Conceptually very similar to Bernard's nice answer, but based on tikz-cd.

\documentclass[12pt]{article}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}[remember picture]
A \arrow[r] & B\\
C \arrow[r] & D\\
\end{tikzcd}
\begin{tikzpicture}[overlay,remember picture]
\path (\tikzcdmatrixname-1-1) to node[midway,sloped]{$\subset$}
(\tikzcdmatrixname-2-1);
\path (\tikzcdmatrixname-1-2) to node[midway,sloped]{$\subset$}
(\tikzcdmatrixname-2-2);
\end{tikzpicture}
\end{document}


Just for fun, an easy solution with the psmatrix environment from pstricks:

 \documentclass[border=5pt, 11pt]{standalone}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{pst-node, auto-pst-pdf}

\newpsobject{ncemptyline}{ncline}{linestyle=none}
\newcommand*\ncsubset[2]{\ncemptyline{#1}{#2}\ncput[nrot=:U]{\subset}}

\begin{document}

$\psset{npos = 0.45} \begin{psmatrix}[rowsep = 0.4, colsep =1cm] [name = L] L & [name = L1] L'\\ [name = K] K & [name = K1] K' \ncsubset{K}{L}\ncsubset{K1}{L1} \psset{arrows=->, arrowinset=0.12, linewidth=0.5pt, nodesep=2pt} \ncline{K}{K1}\ncline{L}{L1} \end{psmatrix}$

\end{document}