When does this converse of Vopěnka's principle hold?

Question

The $n$Lab page on coreflective subcategories cites a theorem of Adamek and Rosický showing that every colimit-closed full subcategory of a locally presentable category is coreflective. My question is, when does the converse hold? If I have a coreflective subcategory which is closed under colimits in the supercategory, how can I tell that it is locally presentable? Does the structure of the inclusion and coreflector tell us anything about this?

Answer 1

This does not look like a converse of your principle, and unless I misunderstood you it seems false.

Indeed, any category is both co-reflective and colimit-closed in itself, but has no reason of being locally presentable.