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    $\begingroup$ This shows what goes wrong if you try to prove that they are equivalent. But it would still be good to see a specific counterexample, as the question asks for! $\endgroup$ Commented Dec 3, 2014 at 16:52
  • $\begingroup$ It would be good to know if $\kappa$-smallness and $\kappa$-compactness really are different. But given that $\aleph_1 \not\triangleleft \aleph_{\omega + 1}$ is basically the smallest non-example of $\kappa \triangleleft \lambda$, it's probably going to be quite intricate... $\endgroup$ Commented Dec 3, 2014 at 17:42
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    $\begingroup$ Thanks Zhen for your answer. Another issue with generalising 3. is that $\lambda$ that appears may not be $\kappa$-filtered. does one face similar difficulty if one tries to prove $\kappa$-smallness is $\gamma$-compactness for some large enough cardinal $\gamma$? $\endgroup$ Commented Dec 4, 2014 at 1:22
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    $\begingroup$ I would think so. But that claim is even harder to disprove because it is vacuous in a locally presentable category. $\endgroup$ Commented Dec 4, 2014 at 7:34