Abstract
Locally presentable categories are precisely those complete categories which have a small dense subcategory. Every subcategory of a locally presentable category (i) has a small dense subcategory, (ii) if it is closed under limits, then it is locally presentable and (iii) if it is closed under colimits, then it is coreflective. For density, canonical colimits can be substituted by arbitrary colimits.
The above results hold under the assumption of the set-theoretical Vopenka's Principle; in fact, each of them is logically equivalent to that principle.
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In Memory of Evelyn Nelson
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Rosický, J., Trnková, V. & Adámek, J. Unexpected properties of locally presentable categories. Algebra Universalis 27, 153–170 (1990). https://doi.org/10.1007/BF01182450
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DOI: https://doi.org/10.1007/BF01182450