-- Lean 4 proof step from Mathlib.CategoryTheory.Yoneda (goal)
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
C : Type u₁
inst_1 : Category.{v₁, u₁} C
Y Z : Cᵒᵖ
f : Y ⟶ Z
F : Cᵒᵖ ⥤ Type (max w v₁)
g : obj (obj (Functor.op uliftYoneda.{w, v₁, u₁} ⋙ coyoneda) Y) F
⊢ Eq (α :=
    obj
      (obj
        (evaluation Cᵒᵖ (Type (max v₁ w)) ⋙
          obj (whiskeringRight (Cᵒᵖ ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
            uliftFunctor.{u₁, max v₁ w})
        Z)
      F)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (map (Functor.op uliftYoneda.{w, v₁, u₁} ⋙ coyoneda) f ≫
            Iso.hom
              ((fun (X : Cᵒᵖ) ↦
                  (NatIso.ofComponents
                      (fun (x : Cᵒᵖ ⥤ Type (max w v₁)) ↦
                        Equiv.toIso (Equiv.trans uliftYonedaEquiv (Equiv.symm Equiv.ulift)))
                      ⋯ :
                    Iso (C := (Cᵒᵖ ⥤ Type (max w v₁)) ⥤ Type (max (max w u₁) v₁))
                      (obj (Functor.op uliftYoneda.{w, v₁, u₁} ⋙ coyoneda) X)
                      (obj
                        (evaluation Cᵒᵖ (Type (max v₁ w)) ⋙
                          obj (whiskeringRight (Cᵒᵖ ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                            uliftFunctor.{u₁, max v₁ w})
                        X)))
                Z))
          F))
      g)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (Iso.hom
              ((fun (X : Cᵒᵖ) ↦
                  (NatIso.ofComponents
                      (fun (x : Cᵒᵖ ⥤ Type (max w v₁)) ↦
                        Equiv.toIso (Equiv.trans uliftYonedaEquiv (Equiv.symm Equiv.ulift)))
                      ⋯ :
                    Iso (C := (Cᵒᵖ ⥤ Type (max w v₁)) ⥤ Type (max (max w u₁) v₁))
                      (obj (Functor.op uliftYoneda.{w, v₁, u₁} ⋙ coyoneda) X)
                      (obj
                        (evaluation Cᵒᵖ (Type (max v₁ w)) ⋙
                          obj (whiskeringRight (Cᵒᵖ ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                            uliftFunctor.{u₁, max v₁ w})
                        X)))
                Y) ≫
            map
              (evaluation Cᵒᵖ (Type (max v₁ w)) ⋙
                obj (whiskeringRight (Cᵒᵖ ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                  uliftFunctor.{u₁, max v₁ w})
              f)
          F))
      g)
have w : Eq (α := obj (obj (evaluation Cᵒᵖ (Type (max v₁ w))) Z) F)
  (ULift.down
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (map (Functor.op uliftYoneda.{w, v₁, u₁} ⋙ coyoneda) f ≫
            Iso.hom
              ((fun (X : Cᵒᵖ) ↦
                  (NatIso.ofComponents
                      (fun (x : Cᵒᵖ ⥤ Type (max w v₁)) ↦
                        Equiv.toIso (Equiv.trans uliftYonedaEquiv (Equiv.symm Equiv.ulift)))
                      sorry :
                    Iso (C := (Cᵒᵖ ⥤ Type (max w v₁)) ⥤ Type (max (max w u₁) v₁))
                      (obj (Functor.op uliftYoneda.{w, v₁, u₁} ⋙ coyoneda) X)
                      (obj
                        (evaluation Cᵒᵖ (Type (max v₁ w)) ⋙
                          obj (whiskeringRight (Cᵒᵖ ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                            uliftFunctor.{u₁, max v₁ w})
                        X)))
                Z))
          F))
      g))
  (ULift.down
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (Iso.hom
              ((fun (X : Cᵒᵖ) ↦
                  (NatIso.ofComponents
                      (fun (x : Cᵒᵖ ⥤ Type (max w v₁)) ↦
                        Equiv.toIso (Equiv.trans uliftYonedaEquiv (Equiv.symm Equiv.ulift)))
                      sorry :
                    Iso (C := (Cᵒᵖ ⥤ Type (max w v₁)) ⥤ Type (max (max w u₁) v₁))
                      (obj (Functor.op uliftYoneda.{w, v₁, u₁} ⋙ coyoneda) X)
                      (obj
                        (evaluation Cᵒᵖ (Type (max v₁ w)) ⋙
                          obj (whiskeringRight (Cᵒᵖ ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                            uliftFunctor.{u₁, max v₁ w})
                        X)))
                Y) ≫
            map
              (evaluation Cᵒᵖ (Type (max v₁ w)) ⋙
                obj (whiskeringRight (Cᵒᵖ ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                  uliftFunctor.{u₁, max v₁ w})
              f)
          F))
      g))
Tactic: rw [← ULift.down_inj]