  Mixing and decoherence are both manifestations of classicality within quantum
theory, each of which admit a very general category-theoretic construction. We
show under which conditions these two 'roads to classicality' coincide. This is
indeed the case for (finite-dimensional) quantum theory, where each
construction yields the category of C*-algebras and completely positive maps.
We present counterexamples where the property fails which includes relational
and modal theories. Finally, we provide a new interpretation for our
category-theoretic generalisation of decoherence in terms of 'leaking
information'.
