∀ {C₁ : Type u_1} {C₂ : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C₁]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁}
  {W₂ : CategoryTheory.MorphismProperty C₂} {Φ : CategoryTheory.LocalizerMorphism W₁ W₂} {X₂ : C₂} {X₁ : C₁}
  (w w_1 : X₂ ⟶ Φ.functor.obj X₁) (e_w : w = w_1) (hw : W₂ w),
  { X₁ := X₁, w := w, hw := hw } = { X₁ := X₁, w := w_1, hw := ⋯ }