CategoryTheory.projective_of_preservesFiniteColimits_preadditiveCoyonedaObj

theorem CategoryTheory.projective_of_preservesFiniteColimits_preadditiveCoyonedaObj : ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] (P : C) [hP : CategoryTheory.Limits.PreservesFiniteColimits (CategoryTheory.preadditiveCoyonedaObj (Opposite.op P))], CategoryTheory.Projective P

This theorem states that for any category `C`, with `C` being an abelian category, if for an object `P` in `C`, its associated preadditive Co-Yoneda functor preserves finite colimits, then the object `P` is projective. In other words, in an abelian category, an object is considered projective if it has the property that the functor derived from it by applying the preadditive Co-Yoneda functor preserves finite colimits.

More concisely: In an abelian category, an object is projective if and only if its preadditive Co-Yoneda functor preserves finite colimits.