structure has not been noticed before.

-   We characterise quantum bijections and quantum isomorphisms as *dagger-dualisable* 1-morphisms (Definition \[def:daggerdualisable\]) in the 2-categories $\QSet$ and $\QGraph$, respectively (Theorem \[thm:dual2\], Theorem \[thm:dualisablequantumgraph\]). We also show that classical bijections and graph isomorphisms are equivalences in these 2-categories (see Proposition \[prop:equivalence\]). In particular, we emphasise that quantum bijections and quantum graph isomorphisms should not be thought of as invertible but merely as dualisable, a characterisation which crucially depends on the $2$-categorical structure.

-   We show that the categories $\QSet(A,B)$ of quantum functions between quantum sets $A$ and $B$, and the categories $\QGraph(G,H)$ of quantum homomorphisms between quantum graphs $G$ and $H$, are *semisimple*, a categorical property which commonly arises in representation theory. This is crucial for the structural understanding of quantum functions, and allows us to characterise certain quantum functions as essentially classical (Definition \[def:classical\]).

Our notions of quantum graph homomorphism and quantum graph isomorphism coincide with those considered in the theory of nonlocal games [@Mancinska2016; @Atserias2016] (Propositions \[prop:mancinskagraphhom\] and \[prop:Atserias\]). We also show that the monoidal category of quantum bijections on a quantum set is the category of finite-dimensional representations of the Hopf $C^*$-algebra associated to Wang’s quantum permutation group (Proposition \[prop:Wang\], [@Wang1998]). 