particular, the category of quantum bijections on a set $X$ quantises the symmetric group $S_X$; as stated above, this is the category of quantum elements of Wang’s quantum permutation group.

A shortcoming of the compact quantum group approach is that it is restricted to automorphisms and cannot easily be applied to study quantum functions or graph homomorphisms between nonisomorphic sets or graphs, as is for example necessary for quantum pseudo-telepathy. In contrast, our approach allows us to consider all quantum functions and quantum graph homomorphisms, including those between nonisomorphic sets and graphs. We show that these form 2-categories $\QSet$ and $\QGraph$, which incorporate all the categories of quantum functions and homomorphisms between different sets and graphs into a single mathematical structure. These $2$-categories unify, generalise, and most importantly expose the connection between work on quantum permutations and quantum graph automorphisms in noncommutative topology, and on quantum graph homomorphisms and isomorphisms in quantum information theory.

Outlook
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We hope that our results lead to the development of further connections between the work on compact quantum groups and quantum information theory. In particular, we expect that the structural understanding of the automorphism categories $\QBij(B,B)$ and $\QIso(G,G)$ can lead to new insights into 