| import Mathlib | |
| open CategoryTheory Limits Functor Opposite | |
| namespace CAT_statement_S_0034 | |
| variable {C : Type u} [Category.{v} C] | |
| def IsFreeObject (U : C ⥤ Type v) (d : C) (I : Type v) : Prop := | |
| ∃ (η : I ⟶ U.obj d), ∀ {y : C} (f : I ⟶ U.obj y), ∃! (g : d ⟶ y), U.map g ∘ η = f | |
| def IsCopower (x d : C) (I : Type v) : Prop := | |
| ∃ (ι : I → (x ⟶ d)), Nonempty (IsColimit (Cofan.mk d ι)) | |
| theorem free_iff_copower_of_representable | |
| (U : C ⥤ Type v) [Faithful U] | |
| (x : C) (hU : U ≅ coyoneda.obj (op x)) | |
| (I : Type v) (d : C) : | |
| IsFreeObject U d I ↔ IsCopower x d I := by | |
| sorry | |
| end CAT_statement_S_0034 | |