  We construct a 2-categorical extension of the relative entropy functor of
Baez and Fritz, and show that our construction is functorial with respect to
vertical morphisms. Moreover, we show such a `2-relative entropy' satisfies
natural 2-categorial analogues of convex linearity, vanishing under optimal
hypotheses, and lower semicontinuity. While relative entropy is a relative
measure of information between probability distributions, we view our
construction as a relative measure of information between channels.
