0.5 ) {$G$};
\draw[string] (0, 4) to (\d,4);
\node[zxvertex=\zxwhite] at (0,4){};
\node[zxvertex=\zxwhite] at (\d,4){};
\node[zxnode=\zxwhite] at (\d/2, 4 ) {$H$};
\draw[string, arrow data={0.08}{>}, arrow data ={0.5}{>},arrow data ={0.95}{>}] (3, -0.5)  to (3,0.5) to [out=up, in=down] (-1.,4) to (-1.,5);
\end{tz}
\end{calign}\\[-15pt]
\end{minipage}
\hspace{\bndr}
&
\begin{minipage}{\wtext}\center quantum \\ graph\\homom. \end{minipage}\\
\hline\nonumber
\begin{minipage}{1.5cm}\center graph \\iso.\end{minipage} &
%%%%%
\begin{minipage}{\wl\textwidth}
\vspace{0.2cm}
\hspace{-0.0cm}bijection $+$
\vspace{-0.1cm}
\begin{calign}\nonumber
\begin{tz}[zx,every to/.style={out=up, in=down},scale=1,xscale=-1]
\draw (0,0) to (0,3);
\path (1.75,0) to (1.75,0.75) to  [looseness=1.1, in looseness=0.9]  node[zxnode=\zxwhite, pos=0.5] {$P$}(-1.75,3);
\node[zxnode=\zxwhite] at (0,0.9){$G$};
\end{tz}
=
\begin{tz}[zx,every to/.style={out=up, in=down},scale=1,xscale=-1]
\draw (0,0) to (0,3);
\path (1.75,0) to (1.75,0.75) to  [looseness=1.1, in looseness=0.9]  node[zxnode=\zxwhite, pos=0.5] {$H$}(-1.75,3);
\node[zxnode=\zxwhite] at (0,0.9){$P$};
\end{tz}
\end{calign}\\[-25pt]
\end{minipage}
\hspace{\bnd}
&
%%%%%%%
\hspace{\bndm}
\begin{minipage}{\wr\textwidth}
\vspace{0.2cm}
\hspace{-0.35cm}quantum bijection $+$
\vspace{-0.1cm}
\begin{calign}\nonumber
\begin{tz}[zx,every to/.style={out=up, in=down},scale=1,xscale=-1]
\draw (0,0) to (0,3);
\draw[arrow data={0.2}{>}, arrow data={0.9}{>}] (1.75,0) to (1.75,0.75) to  [looseness=1.1, in looseness=0.9]  node[zxnode=\zxwhite, pos=0.5] {$P$}(-1.75,3);
\node[zxnode=\zxwhite] at (0,0.9){$G$};
\end{tz}
\quad =\quad 
\begin{tz}[zx,every to/.style={out=up, in=down},scale=1,xscale=-1]
\draw (0,0) to (0,3);
\draw[arrow data={0.2}{>}, arrow data={0.8}{>}] (1.75,0) to [looseness=0.9] node[zxnode=\zxwhite, pos=0.5] {$P$} (-1.75,2.5) to (-1.75,3);
\node[zxnode=\zxwhite] at (0,2.35) {$H$};
\end{tz}
\end{calign}\\[-25pt]
\end{minipage}
\hspace{\bndr}
&
\begin{minipage}{\wtext}\begin{center} quantum \\ graph\\iso.\end{center} \end{minipage}\\ \nonumber&\nonumber&&\\ \hline\end{array}$$

Background {#sec:background}
==========

Gelfand duality
---------------

Our approach to defining a theory of quantum sets and quantum functions is 