  We prove a general categorical theorem that enables us to state that under
certain conditions, the range of a functor is large. As an application, we
prove various results of which the following is a prototype: If every diagram,
indexed by a lattice, of finite Boolean (v,0)-semilattices with
(v,0)-embeddings, can be lifted with respect to the $\Conc$ functor on
lattices, then so can every diagram, indexed by a lattice, of finite
distributive (v,0)-semilattices with (v,0-embeddings. If the premise of this
statement held, this would solve in turn the (still open) problem whether every
distributive algebraic lattice is isomorphic to the congruence lattice of a
lattice. We also outline potential applications of the method to other
functors, such as the $R\mapsto V(R)$ functor on von Neumann regular rings.
