∀ {J : Type u_6} {C : Type u_4} {D : Type u_2} [inst : CategoryTheory.Category.{u_5, u_6} J]
  [inst_1 : CategoryTheory.Category.{u_3, u_4} C] [inst_2 : CategoryTheory.Category.{u_1, u_2} D]
  (G : CategoryTheory.Functor J D) (F : CategoryTheory.Functor C D) [inst_3 : F.Full] [F.Faithful]
  (hG : ∀ (j : J), F.essImage (G.obj j)) (x : J),
  F.map
      (F.preimage
        (CategoryTheory.CategoryStruct.comp (F.toEssImage.objObjPreimageIso { obj := G.obj x, property := ⋯ }).hom.hom
          (CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.CategoryStruct.id x))
            (F.toEssImage.objObjPreimageIso { obj := G.obj x, property := ⋯ }).inv.hom))) =
    F.map (CategoryTheory.CategoryStruct.id (F.toEssImage.objPreimage { obj := G.obj x, property := ⋯ }))