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.Comment: v1: 15 pages. v2: 13 pages, reformatted with minor edits. v3: 15 pages, title changed from "Bootstrapping structural properties, via accessible images," proofs expanded, definitions clarified in response to referees' feedback. v4: 15 pages, dedication added. v5: 15 pages, minor corrections, in press

@article{LiebermanR17,
  title = {Hanf numbers via accessible images},
  author = {Michael Lieberman and Jiri Rosicky},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 13, Issue 2},
  year = {2017},
  doi = {10.23638/lmcs-13(2:11)2017}
}