rough_work.tex

For clarity, let us state the domains and targets of the projection maps explicitly:
\bi{rClcrCl}
\pr_X\cl && X\times Y \to X &\qquad \quad& \pr_{X\times Y}^1 \cl && (X\times Y)\times (X\times Y)\to X\times Y\\
\pr_Y\cl && X\times Y \to Y &\qquad \quad& \pr_{X\times Y}^2 \cl && (X\times Y)\times (X\times Y)\to X\times Y
\ei
Consider the following diagram.
\bse
\begin{tikzcd}[column sep=large,row sep=large]
X\times Y \ar[rr,bend left=30,"\id"] \ar[r,"\Delta"] & \ar[d,"\pr_X\times\pr_Y"](X\times Y)\times(X\times Y)\ar[r,"\pr_{X\times Y}^1"]&X\times Y\ar[d,"\pr_X"]\\
& X\times Y\ar[r,"\pr_X"] & X
\end{tikzcd}
\ese
The top section commutes by definition of $\Delta$, and the square commutes by definition of $\pr_X\times\pr_Y$. We thus have
\bse
\pr_X\circ(\pr_X\times\pr_Y)\circ \Delta=\pr_X\circ\pr_{X\times Y}^1\circ\Delta=\pr_X\circ\id=\pr_X.
\ese
By replacing $\pr_X$ and $\pr_{X\times Y}^1$ in the above diagram with $\pr_Y$ and $\pr_{X\times Y}^2$, respectively, we also obtain
\bse
\pr_Y\circ(\pr_X\times\pr_Y)\circ \Delta=\pr_Y.
\ese
Thus both $(\pr_X\times\pr_Y)\circ \Delta$ and (trivially) $\id$ make the diagram
\bse
\begin{tikzcd}[column sep=large,row sep=large]
&X\times Y\ar[dl,"\pr_X"']\ar[d,dashed]\ar[dr,"\pr_Y"]&\\
X & X\times Y \ar[l,"\pr_X"]\ar[r,"\pr_Y"']& Y
\end{tikzcd}
\ese
commute. But there is only one such map and hence
\bse
(\pr_X\times\pr_Y)\circ \Delta=\id.
\ese






























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    \chapter*{Preface}
    \addcontentsline{toc}{chapter}{Preface}
    \chapter{Looking-Glass House}
\section{A First section}
\section{Another section}
\subsection{A subsection}
    \lipsum[1]
    \chapter{The Garden of Live Flowers}
    \lipsum[2-5]
    \chapter{Looking-Glass Insects}
    \lipsum[6]
    \chapter{Tweedledum and Tweedledee}

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