Barendregt's lambda cube in tikzcd, with labels on arrows

lambda-cube.tex

\documentclass{standalone}
\usepackage{tikz-cd}
\usepackage{amsmath}
\usepackage{tgpagella}

\begin{document}
\begin{tikzcd}
 &  & \lambda\omega \arrow[rrr] &  &  & \lambda\Pi\omega \\
 &  &  &  &  &  \\
\lambda 2 \arrow[rruu] \arrow[rrr] &  &  & \lambda\Pi 2 \arrow[rruu] &  &  \\
 &  & \lambda\underline{\omega} \arrow[rrr] \arrow[uuu] &  &  & \lambda\Pi\underline{\omega} \arrow[uuu] \\
 &  &  &  &  &  \\
\lambda\to
  \arrow{rruu}[sloped,above]{\text{\tiny Types depending on types}}
  \arrow{rruu}[sloped,below]{\text{\tiny Type operators}}
  \arrow{uuu}[sloped,above]{\text{\tiny Values depending on types}}
  \arrow{uuu}[sloped,below]{\text{\tiny Polymorphism}}
  \arrow{rrr}[sloped,above]{\text{\tiny Types depending on values}}
  \arrow{rrr}[sloped,below]{\text{\tiny Dependent types}}
  &  &  & \lambda\Pi \arrow[rruu] \arrow[uuu] &  &
\end{tikzcd}
\end{document}