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    $\begingroup$ Hi James, regarding your "incidentally" parenthetical, yes, the inconsistency of the category-theoretic phrasings of the two statements can be shown without any large-cardinal assumptions: if you has "complete and small dense subcategory implies l.p." along with "$\mathsf{Set}^{\mathrm{op}}$ has a small dense subcategory", then of course $\mathsf{Set}$ would be both l.p. and co-l.p., but that would make $\mathsf{Set}$ a thin category, by a theorem of Gabriel-Ulmer given as 1.64 in Adamek-Rosicky. $\endgroup$ Commented 14 hours ago
  • $\begingroup$ @KevinCarlson Is the inconsistency of (1) and (2) similarly easy to show? $\endgroup$ Commented 14 hours ago
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    $\begingroup$ No, not that I can see. $\mathsf{Set}^{\mathrm{op}}$ has no nontrivial accessibly embedded accesible subcategories at all, nor are any of its accessibility, co-accessibility, well-poweredness, or co-well-poweredness open to foundational question, so it seems rather surprising that these conditions affect each other from a CT viewpoint. $\endgroup$ Commented 12 hours ago