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We assume some basic familiarity with monoidal category theory [@Selinger2010] and $2$-category theory [@Borceux1994 Chapter 7]. Dagger categories are defined in [@Selinger2010]; strict dagger $2$-categories[^5] and dagger $2$-functors are defined in [@Heunen2016]. When we refer to local properties of a 2-functor (such as local faithfulness), we mean that these properties are true of the induced functors on hom-categories. We use the words *module* and *representation* interchangeably. A projector on a Hilbert space $H$ is an endomorphism $P: H \to H$ which is idempotent and self-adjoint $P = P^{\dagger} = P^2$. All sets appearing in this work are finite and, except where clearly specified, all vector spaces and all algebras are finite-dimensional; a notable exception is Section \[sec:universal\]. Consequently, we use the labels  and  for the categories of finite sets and finite-dimensional Hilbert spaces respectively. We also take all $C^*$-algebras to be unital. We denote the $n$-element set by $[n]$.

Acknowledgments {#acknowledgments .unnumbered}
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We are especially grateful to Jamie Vicary for many useful conversations and comments on an earlier draft of this paper. We thank Andre Kornell for a useful conversation about our definitions of quantum functions. We also thank Samson Abramsky, and Rui Soares Barbosa for discussing the 