Colimit-dense subcategories

summary:Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vopěnka's Principle, we prove that a cocomplete category is locally presentable if and only if it has a colimit dense subcategory and a generator consisting of presentable objects. We further show that a $3$-element set is colimit-dense in ${\mathbf{Set}}^{\rm op}$, and spaces of countable dimension are colimit-dense in ${\mathbf{Vec}}^{\rm op}$