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.

\begin{tikzcd}[remember picture]
    A \arrow[r] & B\\
    C \arrow[r] & D\\
\begin{tikzpicture}[overlay,remember picture]
\path (\tikzcdmatrixname-1-1) to node[midway,sloped]{$\subset$}
\path (\tikzcdmatrixname-1-2) to node[midway,sloped]{$\subset$}

enter image description here


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

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



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


enter image description here

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