Likewise, our categories $\QBij([n],[n])$ of quantum bijections on an $n$-element set are the categories of finite-dimensional representations of the Hopf $C^*$-algebra $A(n)$ corresponding to Wang’s quantum permutation group [@Wang1998] (Proposition \[prop:Wang\]), and our categories $\QIso(G,G)$ of quantum automorphisms of a graph $G$ are the categories of finite-dimensional representations of the Hopf $C^*$-algebra $A(G)$ corresponding to Banica’s quantum automorphism group of the graph [@Banica2007]

Our framework therefore captures the finite-dimensional representation theory of the Hopf $C^*$-algebras considered in noncommutative topology. However, we note that a Tannakian correspondence exists only between these algebras and their categories of corepresentations (or, in the language of quantum group theory, with the representations of their associated compact quantum groups [@Neshveyev2013]. Also see Remark \[rem:problemTannaka\] and Remark \[rem:Wangnotdiscrete\]).

Based on the theory of quantum relations developed by Kuperberg and Weaver [@Kuperberg2012; @Weaver2010; @Weaver2015], Kornell [@Kornell2011; @Kornell2018] defines (possibly infinite) quantum sets and quantum functions between them. His notion of finite quantum set coincides with ours; his quantum functions between finite quantum sets are \*-homomorphisms and thus one-dimensional quantum functions in our sense.

#### Quantum information theory.

Quantum graph homomorphisms were defined by Mančinska and Roberson [@Mancinska2016] as generalisations of quantum graph colourings [@Cameron2007] in the context of 