to distinguish between classical and quantum structures.

In Section \[app:quantumrelation\], we show that our reflexive quantum graphs are precisely symmetric and reflexive quantum relations in the sense of Kuperberg and Weaver [@Kuperberg2012; @Weaver2010; @Weaver2015] and fully capture the noncommutative graphs appearing in zero-error communication [@Duan2013].

Related work {#sec:relatedwork}
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A compressed summary of this section can be found in Figure \[fig:relatedwork\].

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#### Quantum symmetry groups and noncommutative topology.

The study of quantum permutation groups — quantum variants of the symmetric groups $S_n$ in noncommutative topology — was suggested by Connes, and carried out by Wang, Banica, Bichon and others [@Wang1998; @Banica2005; @Banica2007; @Banica2007_2; @Bichon2003; @Banica2009; @Banica2007_3]. These quantum permutation groups are compact quantum groups in the sense of Woronowicz [@Woronowicz1998], obtained from a universal construction [@Wang1998]. Quantum automorphism groups of finite graphs are defined similarly [@Bichon2003; @Banica2005]. A more general universal construction of quantum spaces of maps is given by So[ł]{}tan [@Soltan2009] (see Remark \[rem:soltan\]). In Section \[sec:universal\], we show that our categories of quantum functions $\QSet(A,B)$ can be obtained from an analogous universal construction, as the category of finite-dimensional representations of an internal hom $[A,B]$ in the opposite of the category of $C^*$-algebras. 