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Let $C$ be a locally presentable category and let $I$ be a full subcategory of $C$ closed under colimits. Does $I$ have a colimit-dense subset of objects? (I hope that this Q doesn't depend on the chosen set theory.)

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It seems likely that the question is indeed set-theoretical. Under Vopěnka's principle, such a subcategory is automatically accessible (hence, locally presentable). –  Zhen Lin Nov 30 '13 at 0:36
Yes but I wonder if there is another argument. –  Martin Brandenburg Nov 30 '13 at 11:35

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