Representability from a solution set
= Representability from a solution set
On a <complete category> that is <locally small>, a set-valued <functor> is representable exactly when it preserves small <categorical limits> and its <category of elements> has a <weakly initial set>. <Categorical limit> preservation makes that comma <category> complete. The <initial-object lemma for complete categories with a weakly initial set> then supplies a <universal element>, hence a <representation of a functor>.