1.75) to +(0,1.25); at (-0.75,2) [$f$]{}; at (0.75,2) [$f$]{}; at (0,0.75) ;

(0,0) to (0,2) to \[out=45\] (0.75,3); (0,2) to \[out=135\] (-0.75,3); (1.75,0) to \[looseness=0.9\] node\[zxnode=, pos=0.5\] [$f$]{} (-1.75,2.5) to (-1.75,3); at (0,2);

   =   

(0,0) to (0,0.75) to \[out=45\] (0.75,1.75) to (0.75,3); (0,0.75) to \[out=135\] (-0.75,1.75) to (-0.75,3); (1.75,0) to (1.75,0.75) to \[looseness=1.1, in looseness=0.9\] node\[zxnode=, pos=0.36\] [$f$]{} node\[zxnode=, pos=0.64\] [$f$]{}(-1.75,3); at (0,0.75);

The equation on the left is one of the properties of any function between two sets, while the equation on the right is its ‘quantisation’. Further examples of classical morphisms and their quantisations are displayed in Figure \[fig:classicalquantum\].

#### The compositional structure of quantum functions.

Due to the presence of the additional Hilbert space wire introduced in our quantisation procedure, we are naturally led to consider maps on this Hilbert space, which interact in a particular way with quantum functions.

With these additional maps as morphisms, quantum functions between quantum sets $A$ and $B$ are the objects of a category $\QSet(A,B)$. This category of quantum functions between two quantum sets can be understood as a quantisation of the set of classical functions between two classical sets. In other words, our approach to quantisation leads to categorification. In 