-- Module: Mathlib.CategoryTheory.Shift.Quotient
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : Functor.{v, v', u, u'} C D
r : HomRel.{u, v} C
A : Type w
inst_2 : AddMonoid A
inst_3 : HasShift.{v, u, w} C A
inst_4 : HasShift.{v', u', w} D A
inst_5 : r.IsCompatibleWithShift A
inst_6 : F.CommShift A
hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
a b : A
X : C
w :
  Eq (α :=
    (shiftFunctor (Quotient r) b ⋙ lift r F hF).obj ((shiftFunctor C a ⋙ functor r).obj X) ⟶
      (F ⋙ shiftFunctor D (a + b)).obj X)
    (CategoryStruct.comp (Y :=
      (shiftFunctor (Quotient r) b ⋙ lift r F hF).obj ((functor r ⋙ shiftFunctor (Quotient r) a).obj X))
      ((shiftFunctor (Quotient r) b ⋙ lift r F hF).map (NatTrans.app (Iso.hom (Functor.commShiftIso (functor r) a)) X))
      (CategoryStruct.comp (Y :=
        (lift r F hF).obj
          (Functor.obj.{v, v, u, u} (shiftFunctor (Quotient r) b) ((shiftFunctor C a ⋙ functor r).obj X)))
        ((lift r F hF).map
          (Functor.map (shiftFunctor (Quotient r) b) (NatTrans.app (Iso.inv (Functor.commShiftIso (functor r) a)) X)))
        ((lift r F hF).map
            (NatTrans.app (Iso.inv (Functor.commShiftIso (functor r) b))
              (Functor.obj.{v, v, u, u} (shiftFunctor C a) X)) ≫
          F.map (NatTrans.app (Iso.inv (shiftFunctorAdd C a b)) X) ≫
            F.map (NatTrans.app (Iso.hom (shiftFunctorAdd C a b)) X) ≫
              CategoryStruct.comp (Y := (F ⋙ shiftFunctor D b).obj (Functor.obj.{v, v, u, u} (shiftFunctor C a) X))
                (NatTrans.app (Iso.hom (Functor.commShiftIso F b)) (Functor.obj.{v, v, u, u} (shiftFunctor C a) X))
                (Functor.map (shiftFunctor D b) (NatTrans.app (Iso.hom (Functor.commShiftIso F a)) X) ≫
                  NatTrans.app (Iso.inv (shiftFunctorAdd D a b)) (F.obj X)))))
    (CategoryStruct.comp (Y :=
      (shiftFunctor (Quotient r) b ⋙ lift r F hF).obj ((functor r ⋙ shiftFunctor (Quotient r) a).obj X))
      ((shiftFunctor (Quotient r) b ⋙ lift r F hF).map (NatTrans.app (Iso.hom (Functor.commShiftIso (functor r) a)) X))
      (CategoryStruct.comp (Y :=
        (lift r F hF ⋙ shiftFunctor D b).obj
          (Functor.obj.{v, v, u, u} (shiftFunctor (Quotient r) a) ((functor r).obj X)))
        (NatTrans.app (Iso.hom (LiftCommShift.iso F r hF b))
          (Functor.obj.{v, v, u, u} (shiftFunctor (Quotient r) a) ((functor r).obj X)))
        (Functor.map (shiftFunctor D b)
            (CategoryStruct.comp (Y := (lift r F hF).obj ((shiftFunctor C a ⋙ functor r).obj X))
              ((lift r F hF).map (NatTrans.app (Iso.inv (Functor.commShiftIso (functor r) a)) X))
              (NatTrans.app (Iso.hom (Functor.commShiftIso F a)) X)) ≫
          NatTrans.app (Iso.inv (shiftFunctorAdd D a b)) ((lift r F hF).obj ((functor r).obj X)))))
⊢ Eq (α :=
    (lift r F hF).obj
        (Functor.obj.{v, v, u, u} (shiftFunctor (Quotient r) a ⋙ shiftFunctor (Quotient r) b) ((functor r).obj X)) ⟶
      (F ⋙ shiftFunctor D (a + b)).obj X)
    (CategoryStruct.comp (Y :=
      (lift r F hF).obj (Functor.obj.{v, v, u, u} (shiftFunctor (Quotient r) b) ((shiftFunctor C a ⋙ functor r).obj X)))
      ((lift r F hF).map
        (Functor.map (shiftFunctor (Quotient r) b) (NatTrans.app (Iso.inv (Functor.commShiftIso (functor r) a)) X)))
      ((lift r F hF).map
          (NatTrans.app (Iso.inv (Functor.commShiftIso (functor r) b))
            (Functor.obj.{v, v, u, u} (shiftFunctor C a) X)) ≫
        F.map (NatTrans.app (Iso.inv (shiftFunctorAdd C a b)) X) ≫
          F.map (NatTrans.app (Iso.hom (shiftFunctorAdd C a b)) X) ≫
            CategoryStruct.comp (Y := (F ⋙ shiftFunctor D b).obj (Functor.obj.{v, v, u, u} (shiftFunctor C a) X))
              (NatTrans.app (Iso.hom (Functor.commShiftIso F b)) (Functor.obj.{v, v, u, u} (shiftFunctor C a) X))
              (Functor.map (shiftFunctor D b) (NatTrans.app (Iso.hom (Functor.commShiftIso F a)) X) ≫
                NatTrans.app (Iso.inv (shiftFunctorAdd D a b)) (F.obj X))))
    (CategoryStruct.comp (Y :=
      (lift r F hF ⋙ shiftFunctor D b).obj (Functor.obj.{v, v, u, u} (shiftFunctor (Quotient r) a) ((functor r).obj X)))
      (NatTrans.app (Iso.hom (LiftCommShift.iso F r hF b))
        (Functor.obj.{v, v, u, u} (shiftFunctor (Quotient r) a) ((functor r).obj X)))
      (Functor.map (shiftFunctor D b)
          (CategoryStruct.comp (Y := (lift r F hF).obj ((shiftFunctor C a ⋙ functor r).obj X))
            ((lift r F hF).map (NatTrans.app (Iso.inv (Functor.commShiftIso (functor r) a)) X))
            (NatTrans.app (Iso.hom (Functor.commShiftIso F a)) X)) ≫
        NatTrans.app (Iso.inv (shiftFunctorAdd D a b)) ((lift r F hF).obj ((functor r).obj X))))

Tactic:
rw [← cancel_epi ((shiftFunctor (Quotient r) b ⋙ lift r F hF).map
      (NatTrans.app (Functor.commShiftIso (functor r) a).hom X))]

Premises used: CategoryTheory.Quotient, CategoryTheory.Quotient.functor, CategoryTheory.Functor.comp, CategoryTheory.shiftFunctor, CategoryTheory.Functor.map, CategoryTheory.Functor.CommShift.commShiftIso, CategoryTheory.cancel_epi, CategoryTheory.Iso.hom, CategoryTheory.Quotient.lift, CategoryTheory.NatTrans.app