Mathematic schema

Question

Please, can anyone tell me can I draw the following schema in LaTeX?

Answer 1

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 \otimes\cD 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 \otimes\cD Y \\
@VV\delta V @VV\delta V \\
X \star Y @>>> \cD X \star Y + X \star \cD Y \\
@VV\uE V @VV\uE 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.