In the present chapter we will prove that, assuming a large-cardinal axiom of set theory called Vopěnka's principle, the structure of locally presentable categories becomes much more transparent, e.g.,

1. a category is locally presentable iff it is complete and has a dense sub category (Theorem 6.14);

2. each full sub category of a locally presentable category closed under limits (or colimits) is reflective (or coreflective, resp.) (Theorems 6.22, 6.28);

3. each full embedding of an accessible category into a locally presentable category is accessible (Theorem 6.9);

4. every orthogonality class in a locally presentable category is a smallorthogonality class (Corollary 6.24);

5. every subfunctor of an accessible functor is accessible (Corollary 6.31).

In each instance we discuss the reverse implication, i.e., that the property under study implies Vopěnka's principle.

Vopěnka's principle states that no locally presentable category has a large, discrete, full subcategory. We call it a large-cardinal principle because, on the one hand, it implies the existence of measurable cardinals, and on the other hand, its consistency follows from the existence of huge cardinals. We explain this in detail in the Appendix in which all results on large cardinals needed in Chapter 6 are presented. Any reader who refutes the existence of measurable cardinals must refute Vopěnka's principle. However, the present chapter brings a message even to these readers: Do not try to construct counterexamples to the above statements (l)-(5). For example, one might ask whether a given subcategory of RelΣ is reflective. The first thing to check is closedness under limits.