-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
a b : A
X : C
erw :
  CategoryStruct.comp (Y := F'.obj ((shiftFunctor C b ⋙ L).obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
    (F'.map (NatTrans.app (Iso.inv (commShiftIso L b)) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
    ((Lifting.iso L W F F').hom.app
        (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C b) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
      CategoryStruct.comp (Y := (F ⋙ shiftFunctor E b).obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))
        (NatTrans.app (Iso.hom (commShiftIso F b)) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))
        (CategoryStruct.comp (Y :=
          obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
          (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
          (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
            NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X)))))
| CategoryStruct.comp (Y := F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D b) ((L ⋙ shiftFunctor D a).obj X)))
    (F'.map (map (shiftFunctor D b) (NatTrans.app (Iso.hom (commShiftIso L a)) X)))
    (CategoryStruct.comp (Y :=
      obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D a) (L.obj X))))
      (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
        (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D a) (L.obj X)))
      (CategoryStruct.comp (Y :=
        obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
        (map (shiftFunctor E b) (F'.map (NatTrans.app (Iso.inv (commShiftIso L a)) X)))
        (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
          CategoryStruct.comp (Y :=
            obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
            (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
            (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
              NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))))

Tactic:
erw [← NatTrans.naturality_assoc]
      dsimp
      rw [← Functor.map_comp_assoc, ← map_comp, Iso.hom_inv_id_app]
      dsimp
      rw [map_id, map_id, Category.id_comp, commShiftOfLocalization.iso_hom_app,
        Category.assoc, Category.assoc, Category.assoc, ← map_comp_assoc,
        Iso.inv_hom_id_app, map_id, Category.id_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.hom_inv_id_app, CategoryTheory.Functor.map_comp_assoc, CategoryTheory.Functor.commShiftOfLocalization.iso_hom_app, CategoryTheory.Functor.map_id, CategoryTheory.Functor.map_comp, CategoryTheory.Iso.inv_hom_id_app, CategoryTheory.NatTrans.naturality_assoc, CategoryTheory.Category.id_comp