| Summary: | We present several new model-theoretic applications of the fact that, under
the assumption that there exists a proper class of almost strongly compact
cardinals, the powerful image of any accessible functor is accessible. In
particular, we generalize to the context of accessible categories the recent
Hanf number computations of Baldwin and Boney, namely that in an abstract
elementary class (AEC) if the joint embedding and amalgamation properties hold
for models of size up to a sufficiently large cardinal, then they hold for
models of arbitrary size. Moreover, we prove that, under the above-mentioned
large cardinal assumption, every metric AEC is strongly d-tame, strengthening a
result of Boney and Zambrano and pointing the way to further generalizations.
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