Some aspects of the theory of locally presentable categories depends on set-theoretic assumptions. Namely, a large cardinal axiom known as Vopenka's principle implies several nice properties of such categories (and equivalent to some of them). For example, it implies that every cocomplete category with a dense small subcategory is locally presentable.

Now, let $\mathcal{U}$ be some set-theoretic universe such as a Grothendieck universe. Then we can consider the theory of locally presentable $\mathcal{U}$-categories, where, roughly speaking, we replace all small sets with $\mathcal{U}$-small sets. There is a preprint which studies such categories, but it does not consider properties that depend on Vopenka's principle. So, are these properties always true for locally presentable $\mathcal{U}$-categories or do they still depend on set-theoretic assumptions?

Some aspects of the theory of locally presentable categories depends on set-theoretic assumptions. Namely, a large cardinal axiom known as Vopenka's principle implies several nice properties of such categories (and equivalent to some of them). For example, it implies that every cocomplete category with a dense small subcategory is locally presentable.

Now, let $\mathcal{U}$ be some set-theoretic universe such as a Grothendieck universe. Then we can consider the theory of locally presentable $\mathcal{U}$-categories, where, roughly speaking, we replace all small sets with $\mathcal{U}$-small sets. There is a preprint by Zhen Lin Low, Universes for category theory (arXiv:1304.5227), that studies such categories, but it does not consider properties that depend on Vopenka's principle. So, are these properties always true for locally presentable $\mathcal{U}$-categories or do they still depend on set-theoretic assumptions?