-- Lean 4 proof step from Mathlib.CategoryTheory.Shift.Pullback (goal)
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 : PullbackShift.{v_1, u_1, u_2, u_3} C φ
⊢ Eq (α :=
    Quiver.Hom (V := PullbackShift.{v_2, u_4, u_2, u_3} D φ)
      (obj
        (shiftFunctor.{v_1, u_1, u_2} (PullbackShift.{v_1, u_1, u_2, u_3} C φ) (A := A) 0 ⋙ PullbackShift.functor φ F)
        x)
      (obj
        (PullbackShift.functor φ F ⋙ shiftFunctor.{v_2, u_4, u_2} (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (A := A) 0)
        x))
    (NatTrans.app
      (Iso.hom
        (Iso.trans (Y := shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) 0 : B) ⋙ F)
          (isoWhiskerRight (pullbackShiftIso C φ 0 ((φ : (a : A) → B) 0) ⋯) F :
            shiftFunctor.{v_1, u_1, u_2} (PullbackShift.{v_1, u_1, u_2, u_3} C φ) (A := A) 0 ⋙ F ≅
              shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) 0 : B) ⋙ F)
          (Iso.trans (Y := F ⋙ shiftFunctor.{v_2, u_4, u_3} D (A := B) ((φ : (a : A) → B) 0 : B))
              (commShiftIso F ((φ : (a : A) → B) 0))
              (F.isoWhiskerLeft (Iso.symm (pullbackShiftIso D φ 0 ((φ : (a : A) → B) 0) ⋯))) :
            shiftFunctor.{v_1, u_1, u_3} C (A := B) ((φ : (a : A) → B) 0 : B) ⋙ F ≅
              PullbackShift.functor φ F ⋙
                shiftFunctor.{v_2, u_4, u_2} (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (A := A) 0)))
      x)
    (NatTrans.app (Iso.hom (CommShift.isoZero (PullbackShift.functor φ F) A)) x)
have w : Eq (α :=
  Quiver.Hom (V := PullbackShift.{v_2, u_4, u_2, u_3} D φ)
    (obj (PullbackShift.functor φ F)
      (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_2} (PullbackShift.{v_1, u_1, u_2, u_3} C φ) (A := A) 0) x))
    (obj.{v_2, v_2, u_4, u_4} (shiftFunctor.{v_2, u_4, u_2} (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (A := A) 0)
      (obj (PullbackShift.functor φ F) x)))
  (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) 0 : B)) x))
    (F.map (NatTrans.app (Iso.hom (pullbackShiftIso C φ 0 ((φ : (a : A) → B) 0) sorry)) x) :
      F.obj
          (obj.{v_1, v_1, u_1, u_1} (shiftFunctor.{v_1, u_1, u_2} (PullbackShift.{v_1, u_1, u_2, u_3} C φ) (A := A) 0)
            x) ⟶
        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) 0 : B)) x))
    (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) 0 : B)) (F.obj x))
        (NatTrans.app (Iso.hom (commShiftIso F ((φ : (a : A) → B) 0))) x)
        (NatTrans.app (Iso.inv (pullbackShiftIso D φ 0 ((φ : (a : A) → B) 0) sorry)) (F.obj x)) :
      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) 0 : B)) x) ⟶
        obj.{v_2, v_2, u_4, u_4} (shiftFunctor.{v_2, u_4, u_2} (PullbackShift.{v_2, u_4, u_2, u_3} D φ) (A := A) 0)
          (obj (PullbackShift.functor φ F) x)))
  (NatTrans.app (Iso.hom (CommShift.isoZero (PullbackShift.functor φ F) A)) x)
Tactic: dsimp