-- 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
X : C
Y Z : C ⥤ Type (max w v₁)
f : Y ⟶ Z
g : obj (obj (Functor.rightOp uliftCoyoneda.{w, v₁, u₁} ⋙ coyoneda) X) Y
⊢ Eq (α :=
    obj
      (obj
        (evaluation C (Type (max v₁ w)) ⋙
          obj (whiskeringRight (C ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
            uliftFunctor.{u₁, max v₁ w})
        X)
      Z)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (map (obj (Functor.rightOp uliftCoyoneda.{w, v₁, u₁} ⋙ coyoneda) X) f ≫
          Iso.hom
            ((fun (x : C ⥤ Type (max w v₁)) ↦
                (Equiv.toIso (Equiv.trans uliftCoyonedaEquiv (Equiv.symm Equiv.ulift)) :
                  obj (obj (Functor.rightOp uliftCoyoneda.{w, v₁, u₁} ⋙ coyoneda) X) x ≅
                    obj
                      (obj
                        (evaluation C (Type (max v₁ w)) ⋙
                          obj (whiskeringRight (C ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                            uliftFunctor.{u₁, max v₁ w})
                        X)
                      x))
              Z)))
      g)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (Iso.hom
            ((fun (x : C ⥤ Type (max w v₁)) ↦
                (Equiv.toIso (Equiv.trans uliftCoyonedaEquiv (Equiv.symm Equiv.ulift)) :
                  obj (obj (Functor.rightOp uliftCoyoneda.{w, v₁, u₁} ⋙ coyoneda) X) x ≅
                    obj
                      (obj
                        (evaluation C (Type (max v₁ w)) ⋙
                          obj (whiskeringRight (C ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                            uliftFunctor.{u₁, max v₁ w})
                        X)
                      x))
              Y) ≫
          map
            (obj
              (evaluation C (Type (max v₁ w)) ⋙
                obj (whiskeringRight (C ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                  uliftFunctor.{u₁, max v₁ w})
              X)
            f))
      g)
have h : Eq (α := obj (obj (evaluation C (Type (max v₁ w))) X) Z)
  (ULift.down
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (map (obj (Functor.rightOp uliftCoyoneda.{w, v₁, u₁} ⋙ coyoneda) X) f ≫
          Iso.hom
            ((fun (x : C ⥤ Type (max w v₁)) ↦
                (Equiv.toIso (Equiv.trans uliftCoyonedaEquiv (Equiv.symm Equiv.ulift)) :
                  obj (obj (Functor.rightOp uliftCoyoneda.{w, v₁, u₁} ⋙ coyoneda) X) x ≅
                    obj
                      (obj
                        (evaluation C (Type (max v₁ w)) ⋙
                          obj (whiskeringRight (C ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                            uliftFunctor.{u₁, max v₁ w})
                        X)
                      x))
              Z)))
      g))
  (ULift.down
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (Iso.hom
            ((fun (x : C ⥤ Type (max w v₁)) ↦
                (Equiv.toIso (Equiv.trans uliftCoyonedaEquiv (Equiv.symm Equiv.ulift)) :
                  obj (obj (Functor.rightOp uliftCoyoneda.{w, v₁, u₁} ⋙ coyoneda) X) x ≅
                    obj
                      (obj
                        (evaluation C (Type (max v₁ w)) ⋙
                          obj (whiskeringRight (C ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                            uliftFunctor.{u₁, max v₁ w})
                        X)
                      x))
              Y) ≫
          map
            (obj
              (evaluation C (Type (max v₁ w)) ⋙
                obj (whiskeringRight (C ⥤ Type (max v₁ w)) (Type (max v₁ w)) (Type (max (max w u₁) v₁)))
                  uliftFunctor.{u₁, max v₁ w})
              X)
            f))
      g))
Tactic: rw [← ULift.down_inj]