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