  We interpret ontological models for finite-dimensional quantum theory as
functors from the category of finite-dimensional Hilbert spaces and bounded
linear maps to the category of measurable spaces and Markov kernels. This
uniformises several earlier results, that we analyse more closely: Pusey,
Barrett, and Rudolph's result rules out monoidal functors; Leifer and Maroney's
result rules out functors that preserve a duality between states and
measurement; Aaronson et al's result rules out functors that adhere to the
Schr\"odinger equation. We also prove that it is possible to have epistemic
functors that take values in signed Markov kernels.
