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Toy models have been used to separate important features of quantumcomputation from the rich background of the standard Hilbert space model.Category theory, on the other hand, is a general tool to separate components ofmathematical structures, and analyze one layer at a time. It seems natural tocombine the two approaches, and several authors have already pursued this idea.We explore *categorical comprehension construction* as a tool for addingfeatures to toy models. We use it to comprehend quantum propositions andprobabilities within the basic model of finite-dimensional Hilbert spaces. Wealso analyze complementary quantum observables over the category of sets andrelations. This leads into the realm of *test spaces*, a well-studied model. Wepresent one of many possible extensions of this model, enabled by thecomprehension construction. Conspicuously, all models obtained in this waycarry the same categorical structure, *extending* the familiar dagger compactframework with the complementation operations. We call the obtained structure*dagger mix autonomous*, because it extends mix autonomous categories, popularin computer science, in a similar way like dagger compact structure extendscompact categories. Dagger mix autonomous categories seem to arise quitenaturally in quantum computation, as soon as complementarity is viewed as apart of the global structure.

Relating toy models of quantum computation: comprehension, complementarity and dagger mix autonomous categories