  We explore approximate categoricity in the context of distortion systems,
introduced in our previous paper, which are a mild generalization of
perturbation systems, introduced by Ben Yaacov. We extend Ben Yaacov's
Ryll-Nardzewski style characterization of separably approximately categorical
theories from the context of perturbation systems to that of distortion
systems. We also make progress towards an analog of Morley's theorem for
inseparable approximate categoricity, showing that if there is some uncountable
cardinal $\kappa$ such that every model of size $\kappa$ is 'approximately
saturated,' in the appropriate sense, then the same is true for all uncountable
cardinalities. Finally we present some examples of these phenomena and
highlight an apparent interaction between ordinary separable categoricity and
inseparable approximate categoricity.
