Mathematic schema

The diagram in the picture uses the obsolete CD environment of amsmath. OK, thirty-five years ago it was better than nothing!

We can do much better with more modern tools.

documentclass{article}
usepackage{amsmath,amssymb}
usepackage{tikz-cd}

usepackage{amscd} % for emulating the original

newcommand{CC}{mathbb{C}}
newcommand{cP}{mathcal{P}}
newcommand{cD}{mathcal{D}}
newcommand{uE}{mathrm{E}}

begin{document}

This is the diagram obtained with modern tools
begin{equation}
begin{tikzcd}
cP_{CC} otimes cP_{CC} arrow[r,"Delta(D)"] & cP_{CC} otimes cP_{CC}
[-2.5ex] % reduce the vertical space between these two rows
X otimes Y arrow[r] arrow[d,"delta"] & cD X otimes Y + X otimescD Y arrow[d,"delta"]

X star Y arrow[r] arrow[d,"uE"] & cD X star Y + X star cD Y arrow[d,"uE"]

E[X star Y] arrow[r,"cD"] & E[cD X star Y + X star cD Y]
end{tikzcd}
end{equation}

This is instead the original diagram
begin{equation}
begin{CD}
cP_{CC} otimes cP_{CC} @>Delta(D)>> cP_{CC} otimes cP_{CC} 
X otimes Y @>>> cD X otimes Y + X otimescD Y 
@VVdelta V @VVdelta V 
X star Y @>>> cD X star Y + X star cD Y 
@VVuE V @VVuE V 
E[X star Y] @>cD>> E[cD X star Y + X star cD Y]
end{CD}
end{equation}

end{document}

I added also the original for comparison.