-- Module: Mathlib.CategoryTheory.Shift.Pullback
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst_1 : AddMonoid A
inst_2 : AddMonoid B
inst_3 : HasShift.{v_1, u_1, u_3} C B
φ : A →+ B
X : PullbackShift.{v_1, u_1, u_2, u_3} C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = (φ : (a : A) → B) a₁
h₂ : b₂ = (φ : (a : A) → B) a₂
h₃ : b₃ = (φ : (a : A) → B) a₃
D : Type u_4
inst_4 : Category.{v_2, u_4} D
inst_5 : HasShift.{v_2, u_4, u_3} D B
F : Functor.{v_1, v_2, u_1, u_4} C D
inst_6 : F.CommShift B
x x_1 : A
x_2 : PullbackShift.{v_1, u_1, u_2, u_3} C φ
w :
  Eq (α :=
    F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) (x + x_1)) x_2) ⟶
      obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (x + x_1)) (F.obj x_2))
    (CategoryStruct.comp (Y :=
      F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) (x + x_1) : B)) x_2))
      (F.map (NatTrans.app (Iso.hom (pullbackShiftIso C φ (x + x_1) ((φ : (a : A) → B) (x + x_1)) ⋯)) x_2))
      (CategoryStruct.comp (Y :=
        obj.{v_2, v_2, u_4, u_4} (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) (x + x_1) : B))
          (F.obj x_2))
        (NatTrans.app
          (Iso.hom
            (CommShift.isoAdd' ⋯ (commShiftIso F ((φ : (a : A) → B) x)) (commShiftIso F ((φ : (a : A) → B) x_1))))
          x_2)
        (NatTrans.app (Iso.inv (pullbackShiftIso D φ (x + x_1) ((φ : (a : A) → B) (x + x_1)) ⋯)) (F.obj x_2))))
    (CategoryStruct.comp (Y :=
      F.obj
        (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x_1)
          (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)))
      (F.map
        (CategoryStruct.comp (Y :=
          obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) (x + x_1) : B)) x_2)
          (NatTrans.app (Iso.hom (pullbackShiftIso C φ (x + x_1) ((φ : (a : A) → B) (x + x_1)) ⋯)) x_2 :
            obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) (x + x_1)) x_2 ⟶
              obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) (x + x_1) : B)) x_2)
          (CategoryStruct.comp (Y :=
              obj.{v_1, v_1, u_1, u_1}
                (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x : B) ⋙
                  shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x_1 : B))
                x_2)
              (NatTrans.app
                (Iso.hom
                  (shiftFunctorAdd' C ((φ : (a : A) → B) x) ((φ : (a : A) → B) x_1) ((φ : (a : A) → B) (x + x_1)) ⋯))
                x_2)
              (NatTrans.app (Iso.inv (pullbackShiftIso C φ x_1 ((φ : (a : A) → B) x_1) ⋯))
                  (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x : B)) x_2) ≫
                map (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x_1)
                  (NatTrans.app (Iso.inv (pullbackShiftIso C φ x ((φ : (a : A) → B) x) ⋯)) x_2)) :
            obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) (x + x_1) : B)) x_2 ⟶
              obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x_1)
                (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2))))
      (CategoryStruct.comp (Y :=
        F.obj
          (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x_1 : B))
            (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)))
        (F.map
            (NatTrans.app (Iso.hom (pullbackShiftIso C φ x_1 ((φ : (a : A) → B) x_1) ⋯))
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)) :
          F.obj
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x_1)
                (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)) ⟶
            F.obj
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x_1 : B))
                (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)))
        (CategoryStruct.comp (Y :=
            obj.{v_2, v_2, u_4, u_4} (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) x_1 : B))
              (F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)))
            (NatTrans.app (Iso.hom (commShiftIso F ((φ : (a : A) → B) x_1)))
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2))
            (CategoryStruct.comp (Y :=
              obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                (F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)))
              (NatTrans.app (Iso.inv (pullbackShiftIso D φ x_1 ((φ : (a : A) → B) x_1) ⋯))
                (F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)))
              (CategoryStruct.comp (Y :=
                obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                  (F.obj
                    (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x : B)) x_2)))
                (map (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                    (F.map (NatTrans.app (Iso.hom (pullbackShiftIso C φ x ((φ : (a : A) → B) x) ⋯)) x_2)) :
                  obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)) ⟶
                    obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (F.obj
                        (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x : B))
                          x_2)))
                (CategoryStruct.comp (Y :=
                    obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (obj.{v_2, v_2, u_4, u_4} (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) x : B))
                        (F.obj x_2)))
                    (map (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (NatTrans.app (Iso.hom (commShiftIso F ((φ : (a : A) → B) x))) x_2))
                    (CategoryStruct.comp (Y :=
                      obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                        (obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x)
                          (F.obj x_2)))
                      (map (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                        (NatTrans.app (Iso.inv (pullbackShiftIso D φ x ((φ : (a : A) → B) x) ⋯)) (F.obj x_2)))
                      (map (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                          (NatTrans.app (Iso.hom (pullbackShiftIso D φ x ((φ : (a : A) → B) x) ⋯)) (F.obj x_2)) ≫
                        CategoryStruct.comp (Y :=
                          obj.{v_2, v_2, u_4, u_4} (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) x_1 : B))
                            (obj.{v_2, v_2, u_4, u_4}
                              (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) x : B)) (F.obj x_2)))
                          (NatTrans.app (Iso.hom (pullbackShiftIso D φ x_1 ((φ : (a : A) → B) x_1) ⋯))
                            (obj.{v_2, v_2, u_4, u_4}
                              (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) x : B)) (F.obj x_2)))
                          (CategoryStruct.comp (Y :=
                            obj.{v_2, v_2, u_4, u_4}
                              (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) (x + x_1) : B)) (F.obj x_2))
                            (NatTrans.app
                              (Iso.inv
                                (shiftFunctorAdd' D ((φ : (a : A) → B) x) ((φ : (a : A) → B) x_1)
                                  ((φ : (a : A) → B) (x + x_1)) ⋯))
                              (F.obj x_2))
                            (NatTrans.app (Iso.inv (pullbackShiftIso D φ (x + x_1) ((φ : (a : A) → B) (x + x_1)) ⋯))
                              (F.obj x_2))))) :
                  obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (F.obj
                        (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x : B))
                          x_2)) ⟶
                    obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (x + x_1))
                      (F.obj x_2)))) :
          F.obj
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x_1 : B))
                (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)) ⟶
            obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (x + x_1)) (F.obj x_2))))
⊢ Eq (α :=
    F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) (x + x_1)) x_2) ⟶
      obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (x + x_1)) (F.obj x_2))
    (CategoryStruct.comp (Y :=
      F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) (x + x_1) : B)) x_2))
      (F.map (NatTrans.app (Iso.hom (pullbackShiftIso C φ (x + x_1) ((φ : (a : A) → B) (x + x_1)) ⋯)) x_2) :
        F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) (x + x_1)) x_2) ⟶
          F.obj
            (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) (x + x_1) : B)) x_2))
      (CategoryStruct.comp (Y :=
          obj.{v_2, v_2, u_4, u_4} (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) (x + x_1) : B))
            (F.obj x_2))
          (NatTrans.app (Iso.hom (commShiftIso F ((φ : (a : A) → B) (x + x_1)))) x_2)
          (NatTrans.app (Iso.inv (pullbackShiftIso D φ (x + x_1) ((φ : (a : A) → B) (x + x_1)) ⋯)) (F.obj x_2)) :
        F.obj
            (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) (x + x_1) : B)) x_2) ⟶
          obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (x + x_1)) (F.obj x_2)))
    (F.map (NatTrans.app (Iso.hom (shiftFunctorAdd (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x x_1)) x_2) ≫
      CategoryStruct.comp (Y :=
        F.obj
          (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x_1 : B))
            (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)))
        (F.map
            (NatTrans.app (Iso.hom (pullbackShiftIso C φ x_1 ((φ : (a : A) → B) x_1) ⋯))
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)) :
          F.obj
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x_1)
                (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)) ⟶
            F.obj
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x_1 : B))
                (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)))
        (CategoryStruct.comp (Y :=
            obj.{v_2, v_2, u_4, u_4} (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) x_1 : B))
              (F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)))
            (NatTrans.app (Iso.hom (commShiftIso F ((φ : (a : A) → B) x_1)))
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2))
            (NatTrans.app (Iso.inv (pullbackShiftIso D φ x_1 ((φ : (a : A) → B) x_1) ⋯))
                (F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)) ≫
              CategoryStruct.comp (Y :=
                obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                  (F.obj
                    (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x : B)) x_2)))
                (map (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                    (F.map (NatTrans.app (Iso.hom (pullbackShiftIso C φ x ((φ : (a : A) → B) x) ⋯)) x_2)) :
                  obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (F.obj (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)) ⟶
                    obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (F.obj
                        (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x : B))
                          x_2)))
                (CategoryStruct.comp (Y :=
                    obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (obj.{v_2, v_2, u_4, u_4} (shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) x : B))
                        (F.obj x_2)))
                    (map (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (NatTrans.app (Iso.hom (commShiftIso F ((φ : (a : A) → B) x))) x_2))
                    (map (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                        (NatTrans.app (Iso.inv (pullbackShiftIso D φ x ((φ : (a : A) → B) x) ⋯)) (F.obj x_2)) ≫
                      NatTrans.app (Iso.inv (shiftFunctorAdd (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x x_1))
                        (F.obj x_2)) :
                  obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) x_1)
                      (F.obj
                        (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x : B))
                          x_2)) ⟶
                    obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (x + x_1))
                      (F.obj x_2))) :
          F.obj
              (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) x_1 : B))
                (obj.{v_1, v_1, u_1, u_1} (shiftFunctor (PullbackShift.{v_1, u_1, u_2, u_3} C φ) x) x_2)) ⟶
            obj.{v_2, v_2, u_4, u_4} (shiftFunctor (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (x + x_1)) (F.obj x_2)))

Tactic:
rw [F.commShiftIso_add' (φ.map_add _ _).symm,
      ← shiftFunctorAdd'_eq_shiftFunctorAdd, ← shiftFunctorAdd'_eq_shiftFunctorAdd,
      pullbackShiftFunctorAdd'_hom_app φ _ _ _ _ rfl _ _ _ rfl rfl rfl,
      pullbackShiftFunctorAdd'_inv_app φ _ _ _ _ rfl _ _ _ rfl rfl rfl]

Premises used: CategoryTheory.Functor.commShiftIso_add', CategoryTheory.pullbackShiftFunctorAdd'_hom_app, Eq.symm, AddMonoidHom.map_add, CategoryTheory.pullbackShiftFunctorAdd'_inv_app, rfl, CategoryTheory.shiftFunctorAdd'_eq_shiftFunctorAdd