duality between finite sets and finite-dimensional commutative $C^*$-algebras in this setting.

In Section \[sec:quantumset\], we introduce quantum sets, quantum elements and quantum functions, and define the 2-category $\QSet$ which encodes their compositional structure. We justify our definitions by a universal property.

In Section \[sec:quantumbijection\], we quantise bijections and prove that the resulting quantum bijections are precisely dagger-dualisable 1-morphisms in $\QSet$. We show that quantum bijections between classical sets correspond to projective permutation matrices as defined in the theory of nonlocal games [@Atserias2016], and to finite-dimensional representations of Wang’s quantum symmetry group algebras [@Wang1998].

In Section \[sec:qgt\], we quantise finite graphs and their homomorphisms and define the $\QGraph$ which encodes their compositional structure. We show that our definitions capture the quantum graph homomorphisms and isomorphisms of Man[č]{}inska and Roberson [@Mancinska2016] and Atserias et al [@Atserias2016], as well as the finite-dimensional representation theory of Banica and Bichon’s quantum automorphism group algebras [@Banica2005; @Bichon2003]. We also show that quantum graph isomorphisms are precisely dagger-dualisable 1-morphisms in $\QGraph$.

In Section \[sec:semisimple\], we show that the categories of quantum functions and quantum graph homomorphisms are semisimple. We discuss the operational interpretation of the direct sum of quantum functions and show how this can be used 