Uncountable categoricity of local abstract elementary classes with amalgamation

We give a complete and elementary proof of the following upward categoricity theorem: let K be a local abstract elementary class with amalgamation and joint embedding, arbitrarily large models, and countable Löwenheim–Skolem number. If K is categorical in ℵ1 then K is categorical in every uncountable cardinal. In particular, this provides a new proof of the upward part of Morley’s theorem in first order logic without any use of prime models or heavy stability theoretic machinery (dependence relations, Morley rank, etc.).