used to perform a task classically impossible without communication. Such tasks are usually formulated as games, where winning classical strategies correspond to certain homomorphisms between combinatorial structures; quantum strategies can then be understood as quantised versions of these homomorphisms. One such game is the *graph homomorphism game* [@Mancinska2016], leading to a notion of *quantum graph homomorphism*. We show that quantum graph homomorphisms are quantum functions between vertex sets preserving the graph structure; in fact, quantum functions can themselves be understood as perfect quantum strategies for a certain ‘function game’. The theory of quantum functions places this game-theoretic approach to quantisation into a broader mathematical context, in particular relating it to more conventional approaches to quantisation.=-2

Indeed, since the advent of quantum mechanics, quantisation has been associated with the passage from commuting to noncommuting variables. In the spirit of noncommutative topology[^2], we regard $C^*$-algebras as noncommutative analogues of topological spaces, or *quantum spaces*, with finite-dimensional $C^*$-algebras playing the role of *finite quantum sets*. Finding an appropriate definition of a quantum space of quantum functions between finite sets is a more subtle issue. Conventionally, these quantum spaces are defined as $C^*$-algebras satisfying a universal property. A prominent example is Wang’s *quantum permutation 