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