-- Module: Mathlib.CategoryTheory.DifferentialObject
Goal:
S : Type u_1
inst : AddCommGroupWithOne S
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasZeroMorphisms.{v, u} C
inst_3 : HasShift.{v, u, u_1} C S
m n : S
X : DifferentialObject.{v, u, u_1} S C
refine_1 :
  Eq (α :=
    Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m ⋙ CategoryTheory.shiftFunctor C n) X.obj ⟶
      Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor.{v, u, u_1} C (A := S) 1)
        (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C n)
          (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m) X.obj)))
    (NatTrans.app (Iso.inv (CategoryTheory.shiftFunctorAdd C m n)) X.obj ≫
      Functor.map (CategoryTheory.shiftFunctor C (m + n)) X.d ≫
        NatTrans.app (Iso.hom (shiftFunctorComm.{v, u, u_1} C 1 (m + n))) X.obj ≫
          Functor.map (CategoryTheory.shiftFunctor.{v, u, u_1} C (A := S) 1)
            (NatTrans.app (Iso.hom (CategoryTheory.shiftFunctorAdd C m n)) X.obj))
    (CategoryStruct.comp (Y :=
      Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C n)
        (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor.{v, u, u_1} C (A := S) 1)
          (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m) X.obj)))
      (Functor.map (CategoryTheory.shiftFunctor C n)
        (Functor.map (CategoryTheory.shiftFunctor C m) X.d ≫
          NatTrans.app (Iso.hom (shiftFunctorComm.{v, u, u_1} C 1 m)) X.obj))
      (NatTrans.app (Iso.hom (shiftFunctorComm.{v, u, u_1} C 1 n))
        (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m) X.obj)))
⊢ Eq (α :=
    Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m ⋙ CategoryTheory.shiftFunctor C n) X.obj ⟶
      Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor.{v, u, u_1} C (A := S) 1)
        (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C n)
          (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m) X.obj)))
    (NatTrans.app (Iso.inv (CategoryTheory.shiftFunctorAdd C m n)) X.obj ≫
        (Functor.map (CategoryTheory.shiftFunctor C (m + n)) X.d ≫
            NatTrans.app (Iso.hom (shiftFunctorComm.{v, u, u_1} C 1 (m + n))) X.obj) ≫
          Functor.map (CategoryTheory.shiftFunctor.{v, u, u_1} C (A := S) 1)
            (NatTrans.app (Iso.hom (CategoryTheory.shiftFunctorAdd C m n)) X.obj) :
      Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m ⋙ CategoryTheory.shiftFunctor C n) X.obj ⟶
        Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor.{v, u, u_1} C (A := S) 1)
          (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C n)
            (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m) X.obj)))
    (NatTrans.app (Iso.inv (CategoryTheory.shiftFunctorAdd C m n)) X.obj ≫
        NatTrans.app (Iso.hom (CategoryTheory.shiftFunctorAdd C m n)) X.obj ≫
          CategoryStruct.comp (Y :=
            Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C n)
              (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor.{v, u, u_1} C (A := S) 1)
                (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m) X.obj)))
            (Functor.map (CategoryTheory.shiftFunctor C n)
              (Functor.map (CategoryTheory.shiftFunctor C m) X.d ≫
                NatTrans.app (Iso.hom (shiftFunctorComm.{v, u, u_1} C 1 m)) X.obj))
            (NatTrans.app (Iso.hom (shiftFunctorComm.{v, u, u_1} C 1 n))
              (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m) X.obj)) :
      Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m ⋙ CategoryTheory.shiftFunctor C n) X.obj ⟶
        Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor.{v, u, u_1} C (A := S) 1)
          (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C n)
            (Functor.obj.{v, v, u, u} (CategoryTheory.shiftFunctor C m) X.obj)))

Tactic:
simp only [Category.assoc, Iso.inv_hom_id_app_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.inv_hom_id_app_assoc