group* [@Wang1998] of a set, defined in the framework of compact quantum groups [@Woronowicz1998] and later extended to the *quantum automorphism group* [@Banica2005] of a graph. Our framework captures the finite-dimensional representation theory of these $C^*$-algebras and extends the existing theory of quantum permutation and automorphism groups to a theory of general morphisms between different sets and graphs. In particular, it relates quantum automorphism groups to the *quantum graph isomorphisms* [@Atserias2016] considered in the study of pseudo-telepathy, providing a concrete link between results in noncommutative topology and recent problems in quantum information theory.

Our theory of quantum functions — unifying and extending the above approaches to the quantisation of combinatorial structures — is formulated in categorical terms and allows us to apply techniques from categorical algebra, such as the notions of semisimplicity and dualisability. The following is a brief summary of our main results:

-   We show that quantum sets and quantum functions naturally form a 2-category $\QSet$ (Definition \[def:2catqset\]) ‘quantising’ the category of finite sets and functions. Similarly, quantum graphs (Definition \[def:quantumgraphsbyadjmats\], see also [@Duan2013; @Weaver2015]) and quantum graph homomorphisms naturally form a 2-category $\QGraph$ (Definition \[def:2catqgraph\]) ‘quantising’ the category of finite graphs and graph homomorphisms. This higher compositional 