-- Module: Mathlib.CategoryTheory.Shift.CommShift
Goal:
C : Type u_1
D : Type u_2
E : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Category.{v_3, u_3} E
F : Functor.{v_1, v_2, u_1, u_2} C D
G : Functor.{v_2, v_3, u_2, u_3} D E
A : Type u_4
B : Type u_5
inst_3 : AddMonoid A
inst_4 : AddCommMonoid B
inst_5 : HasShift.{v_1, u_1, u_4} C A
inst_6 : HasShift.{v_2, u_2, u_4} D A
inst_7 : HasShift.{v_3, u_3, u_4} E A
inst_8 : HasShift.{v_1, u_1, u_5} C B
inst_9 : HasShift.{v_2, u_2, u_5} D B
inst_10 : F.CommShift A
inst_11 : G.CommShift A
a b : A
X : C
w :
  𝟙 (G.obj (F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (a + b)) X))) ≫
      G.map
          (F.map (NatTrans.app (Iso.hom (shiftFunctorAdd C a b)) X) ≫
            CategoryStruct.comp (Y := (F ⋙ shiftFunctor D b).obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) X))
              (NatTrans.app (Iso.hom (commShiftIso F b)) (obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) X))
              (map (shiftFunctor D b) (NatTrans.app (Iso.hom (commShiftIso F a)) X) ≫
                NatTrans.app (Iso.inv (shiftFunctorAdd D a b)) (F.obj X))) ≫
        𝟙 (G.obj (obj.{v_2, v_2, u_2, u_2} (shiftFunctor D (a + b)) (F.obj X))) ≫
          (G.map (NatTrans.app (Iso.hom (shiftFunctorAdd D a b)) (F.obj X)) ≫
              CategoryStruct.comp (Y :=
                (G ⋙ shiftFunctor E b).obj (obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) (F.obj X)))
                (NatTrans.app (Iso.hom (commShiftIso G b)) (obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) (F.obj X)))
                (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso G a)) (F.obj X)) ≫
                  NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (G.obj (F.obj X)))) ≫
            𝟙 (obj.{v_3, v_3, u_3, u_3} (shiftFunctor E (a + b)) (G.obj (F.obj X))) =
    (F ⋙ G).map (NatTrans.app (Iso.hom (shiftFunctorAdd C a b)) X) ≫
      NatTrans.app
          (Iso.hom
            (Iso.symm (associator (shiftFunctor C b) F G) ≪≫
              isoWhiskerRight (commShiftIso F b) G ≪≫
                F.associator (shiftFunctor D b) G ≪≫
                  F.isoWhiskerLeft (commShiftIso G b) ≪≫ (F.associator G (shiftFunctor E b)).symm))
          (obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) X) ≫
        map (shiftFunctor E b)
            (NatTrans.app
              (Iso.hom
                (Iso.symm (associator (shiftFunctor C a) F G) ≪≫
                  isoWhiskerRight (commShiftIso F a) G ≪≫
                    F.associator (shiftFunctor D a) G ≪≫
                      F.isoWhiskerLeft (commShiftIso G a) ≪≫ (F.associator G (shiftFunctor E a)).symm))
              X) ≫
          NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) ((F ⋙ G).obj X)
⊢ 𝟙 (G.obj (F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (a + b)) X))) ≫
      G.map (NatTrans.app (Iso.hom (commShiftIso F (a + b))) X) ≫
        𝟙 (G.obj (obj.{v_2, v_2, u_2, u_2} (shiftFunctor D (a + b)) (F.obj X))) ≫
          NatTrans.app (Iso.hom (commShiftIso G (a + b))) (F.obj X) ≫
            𝟙 (obj.{v_3, v_3, u_3, u_3} (shiftFunctor E (a + b)) (G.obj (F.obj X))) =
    NatTrans.app
      (Iso.hom
        (isoAdd
          (Iso.symm (associator (shiftFunctor C a) F G) ≪≫
            isoWhiskerRight (commShiftIso F a) G ≪≫
              F.associator (shiftFunctor D a) G ≪≫
                F.isoWhiskerLeft (commShiftIso G a) ≪≫ (F.associator G (shiftFunctor E a)).symm)
          (Iso.symm (associator (shiftFunctor C b) F G) ≪≫
            isoWhiskerRight (commShiftIso F b) G ≪≫
              F.associator (shiftFunctor D b) G ≪≫
                F.isoWhiskerLeft (commShiftIso G b) ≪≫ (F.associator G (shiftFunctor E b)).symm)))
      X

Tactic:
simp only [commShiftIso_add, isoAdd_hom_app]

Premises used: CategoryTheory.Functor.CommShift.isoAdd_hom_app, CategoryTheory.Functor.CommShift.commShiftIso_add