-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
Y : C
n₀ n₁ : ℕ
h : 1 + n₀ = n₁
this_1 : HasDerivedCategory C := HasDerivedCategory.standard C
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : Ext.{w, v, u} S.X₁ Y n₀
⊢ Eq (α :=
    ShiftedHom (C := DerivedCategory C) (M := ℤ) (Functor.obj (singleFunctor C 0) S.X₃)
      (Functor.obj (singleFunctor C 0) Y) (↑n₁ : ℤ))
    (((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v)))
                (CC := AddCommGrpCat.carrier)
                (ShortComplex.mk (X₁ :=
                    Functor.obj (C := (DerivedCategory C)ᵒᵖ)
                      (Functor.shift (Functor.obj preadditiveYoneda (Functor.obj (singleFunctor C 0) Y)) (M := ℤ)
                        (↑n₀ : ℤ))
                      (Triangle.obj₃
                        (Functor.obj (Equivalence.functor (triangleOpEquivalence (DerivedCategory C)))
                          (op hS.singleTriangle))))
                    (Functor.homologySequenceδ (Functor.obj preadditiveYoneda (Functor.obj (singleFunctor C 0) Y))
                      (Functor.obj (Equivalence.functor (triangleOpEquivalence (DerivedCategory C)))
                        (op hS.singleTriangle))
                      (↑n₀ : ℤ) (↑n₁ : ℤ) ⋯)
                    (Functor.map
                      (Functor.shift (Functor.obj preadditiveYoneda (Functor.obj (singleFunctor C 0) Y)) (M := ℤ)
                        (↑n₁ : ℤ))
                      (Triangle.mor₁
                        (Functor.obj (Equivalence.functor (triangleOpEquivalence (DerivedCategory C)))
                          (op hS.singleTriangle))))
                    ⋯).f).comp
            (↑homAddEquiv :
              Ext S.X₁ Y n₀ →+ ShiftedHom ((singleFunctor C 0).obj S.X₁) ((singleFunctor C 0).obj Y) (↑n₀ : ℤ)) :
          (a : Ext S.X₁ Y n₀) → ShiftedHom ((singleFunctor C 0).obj S.X₃) ((singleFunctor C 0).obj Y) (↑n₁ : ℤ))
        x :
      ShiftedHom (C := DerivedCategory C) (M := ℤ) (Functor.obj (singleFunctor C 0) S.X₃)
        (Functor.obj (singleFunctor C 0) Y) (↑n₁ : ℤ))
    ((AddMonoidHom.comp
            (↑homAddEquiv :
              Ext S.X₃ Y n₁ →+ ShiftedHom ((singleFunctor C 0).obj S.X₃) ((singleFunctor C 0).obj Y) (↑n₁ : ℤ))
            (ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w)) (CC :=
              AddCommGrpCat.carrier)
              (ShortComplex.mk (X₁ := AddCommGrpCat.of (Ext.{w, v, u} S.X₁ Y n₀))
                  (AddCommGrpCat.ofHom (precomp hS.extClass Y h)) (AddCommGrpCat.ofHom (precomp (mk₀ S.g) Y ⋯)) ⋯).f) :
          (a : Ext S.X₁ Y n₀) → ShiftedHom ((singleFunctor C 0).obj S.X₃) ((singleFunctor C 0).obj Y) (↑n₁ : ℤ))
        x :
      ShiftedHom (C := DerivedCategory C) (M := ℤ) (Functor.obj (singleFunctor C 0) S.X₃)
        (Functor.obj (singleFunctor C 0) Y) (↑n₁ : ℤ))
have comm₁₂ : Eq (α :=
  ShiftedHom (C := DerivedCategory C) (M := ℤ) (Functor.obj (singleFunctor C 0) S.X₃)
    (Functor.obj (singleFunctor C 0) Y) (↑n₁ : ℤ))
  (((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CC :=
            AddCommGrpCat.carrier) (X :=
            Functor.obj
              (Functor.shift
                (preadditiveYonedaObj (Functor.obj (singleFunctor C 0) Y) ⋙
                  forget₂ (ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) (FC :=
                    fun (x1 x2 : ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) ↦
                    LinearMap (R := End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))
                      (RingHom.id (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y)))
                      (↑x1 : Type (max u v)) (↑x2 : Type (max u v)))
                    (CC := ModuleCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat) ↦
                    (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddCommGrpCat.carrier))
                (M := ℤ) (↑n₀ : ℤ))
              (op (Functor.obj (singleFunctor C 0) S.X₁)))
            (Y :=
            Functor.obj
              (Functor.shift
                (preadditiveYonedaObj (Functor.obj (singleFunctor C 0) Y) ⋙
                  forget₂ (ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) (FC :=
                    fun (x1 x2 : ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) ↦
                    LinearMap (R := End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))
                      (RingHom.id (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y)))
                      (↑x1 : Type (max u v)) (↑x2 : Type (max u v)))
                    (CC := ModuleCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat) ↦
                    (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddCommGrpCat.carrier))
                (M := ℤ) (↑n₁ : ℤ))
              (op (Functor.obj (singleFunctor C 0) S.X₃)))
            (Functor.homologySequenceδ
                (preadditiveYonedaObj (Functor.obj (singleFunctor C 0) Y) ⋙
                  forget₂ (ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) (FC :=
                    fun (x1 x2 : ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) ↦
                    LinearMap (R := End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))
                      (RingHom.id (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y)))
                      (↑x1 : Type (max u v)) (↑x2 : Type (max u v)))
                    (CC := ModuleCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat) ↦
                    (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddCommGrpCat.carrier))
                (Triangle.mk (X := op (Functor.obj (singleFunctor C 0) S.X₃))
                  (Quiver.Hom.op (Functor.map (singleFunctor C 0) S.g))
                  (Quiver.Hom.op (Functor.map (singleFunctor C 0) S.f))
                  (CategoryStruct.comp (Y :=
                    Functor.obj (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                      (op
                        (Functor.obj (shiftFunctor (DerivedCategory C) (A := ℤ) 1)
                          (Functor.obj (singleFunctor C 0) S.X₁))))
                    (NatTrans.app (Iso.inv (Equivalence.counitIso (opShiftFunctorEquivalence (DerivedCategory C) 1)))
                        (op (Functor.obj (singleFunctor C 0) S.X₁)) :
                      Quiver.Hom (V := (DerivedCategory C)ᵒᵖ)
                        (Functor.obj (𝟭 (DerivedCategory C)ᵒᵖ) (op (Functor.obj (singleFunctor C 0) S.X₁)))
                        (Functor.obj
                          (Functor.op (shiftFunctor (DerivedCategory C) (A := ℤ) 1) ⋙
                            shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                          (op (Functor.obj (singleFunctor C 0) S.X₁))))
                    (Functor.map (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1) (Quiver.Hom.op hS.singleδ) :
                      Quiver.Hom (V := (DerivedCategory C)ᵒᵖ)
                        (Functor.obj (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                          (op
                            (Functor.obj (shiftFunctor (DerivedCategory C) (A := ℤ) 1)
                              (Functor.obj (singleFunctor C 0) S.X₁))))
                        (Functor.obj (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                          (op (Functor.obj (singleFunctor C 0) S.X₃))))))
                (↑n₀ : ℤ) (↑n₁ : ℤ) sorry :
              Functor.obj
                  (Functor.shift
                    (preadditiveYonedaObj (Functor.obj (singleFunctor C 0) Y) ⋙
                      forget₂ (ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) (FC :=
                        fun (x1 x2 : ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) ↦
                        LinearMap (R := End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))
                          (RingHom.id (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y)))
                          (↑x1 : Type (max u v)) (↑x2 : Type (max u v)))
                        (CC := ModuleCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat) ↦
                        (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddCommGrpCat.carrier))
                    (M := ℤ) (↑n₀ : ℤ))
                  (Triangle.obj₃
                    (Triangle.mk (X := op (Functor.obj (singleFunctor C 0) S.X₃))
                      (Quiver.Hom.op (Functor.map (singleFunctor C 0) S.g))
                      (Quiver.Hom.op (Functor.map (singleFunctor C 0) S.f))
                      (CategoryStruct.comp (Y :=
                        Functor.obj (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                          (op
                            (Functor.obj (shiftFunctor (DerivedCategory C) (A := ℤ) 1)
                              (Functor.obj (singleFunctor C 0) S.X₁))))
                        (NatTrans.app
                            (Iso.inv (Equivalence.counitIso (opShiftFunctorEquivalence (DerivedCategory C) 1)))
                            (op (Functor.obj (singleFunctor C 0) S.X₁)) :
                          Quiver.Hom (V := (DerivedCategory C)ᵒᵖ)
                            (Functor.obj (𝟭 (DerivedCategory C)ᵒᵖ) (op (Functor.obj (singleFunctor C 0) S.X₁)))
                            (Functor.obj
                              (Functor.op (shiftFunctor (DerivedCategory C) (A := ℤ) 1) ⋙
                                shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                              (op (Functor.obj (singleFunctor C 0) S.X₁))))
                        (Functor.map (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1) (Quiver.Hom.op hS.singleδ) :
                          Quiver.Hom (V := (DerivedCategory C)ᵒᵖ)
                            (Functor.obj (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                              (op
                                (Functor.obj (shiftFunctor (DerivedCategory C) (A := ℤ) 1)
                                  (Functor.obj (singleFunctor C 0) S.X₁))))
                            (Functor.obj (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                              (op (Functor.obj (singleFunctor C 0) S.X₃))))))) ⟶
                Functor.obj
                  (Functor.shift
                    (preadditiveYonedaObj (Functor.obj (singleFunctor C 0) Y) ⋙
                      forget₂ (ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) (FC :=
                        fun (x1 x2 : ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) ↦
                        LinearMap (R := End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))
                          (RingHom.id (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y)))
                          (↑x1 : Type (max u v)) (↑x2 : Type (max u v)))
                        (CC := ModuleCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat) ↦
                        (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddCommGrpCat.carrier))
                    (M := ℤ) (↑n₁ : ℤ))
                  (Triangle.obj₁
                    (Triangle.mk (X := op (Functor.obj (singleFunctor C 0) S.X₃))
                      (Quiver.Hom.op (Functor.map (singleFunctor C 0) S.g))
                      (Quiver.Hom.op (Functor.map (singleFunctor C 0) S.f))
                      (CategoryStruct.comp (Y :=
                        Functor.obj (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                          (op
                            (Functor.obj (shiftFunctor (DerivedCategory C) (A := ℤ) 1)
                              (Functor.obj (singleFunctor C 0) S.X₁))))
                        (NatTrans.app
                            (Iso.inv (Equivalence.counitIso (opShiftFunctorEquivalence (DerivedCategory C) 1)))
                            (op (Functor.obj (singleFunctor C 0) S.X₁)) :
                          Quiver.Hom (V := (DerivedCategory C)ᵒᵖ)
                            (Functor.obj (𝟭 (DerivedCategory C)ᵒᵖ) (op (Functor.obj (singleFunctor C 0) S.X₁)))
                            (Functor.obj
                              (Functor.op (shiftFunctor (DerivedCategory C) (A := ℤ) 1) ⋙
                                shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                              (op (Functor.obj (singleFunctor C 0) S.X₁))))
                        (Functor.map (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1) (Quiver.Hom.op hS.singleδ) :
                          Quiver.Hom (V := (DerivedCategory C)ᵒᵖ)
                            (Functor.obj (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                              (op
                                (Functor.obj (shiftFunctor (DerivedCategory C) (A := ℤ) 1)
                                  (Functor.obj (singleFunctor C 0) S.X₁))))
                            (Functor.obj (shiftFunctor (DerivedCategory C)ᵒᵖ (A := ℤ) 1)
                              (op (Functor.obj (singleFunctor C 0) S.X₃)))))))) :
          (↑(Functor.obj
                  (Functor.shift
                    (preadditiveYonedaObj (Functor.obj (singleFunctor C 0) Y) ⋙
                      forget₂ (ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) (FC :=
                        fun (x1 x2 : ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) ↦
                        LinearMap (R := End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))
                          (RingHom.id (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y)))
                          (↑x1 : Type (max u v)) (↑x2 : Type (max u v)))
                        (CC := ModuleCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat) ↦
                        (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddCommGrpCat.carrier))
                    (M := ℤ) (↑n₀ : ℤ))
                  (op (Functor.obj (singleFunctor C 0) S.X₁))) :
              Type (max u v)) →+
            (↑(Functor.obj
                  (Functor.shift
                    (preadditiveYonedaObj (Functor.obj (singleFunctor C 0) Y) ⋙
                      forget₂ (ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) (FC :=
                        fun (x1 x2 : ModuleCat (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))) ↦
                        LinearMap (R := End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y))
                          (RingHom.id (End (C := DerivedCategory C) (Functor.obj (singleFunctor C 0) Y)))
                          (↑x1 : Type (max u v)) (↑x2 : Type (max u v)))
                        (CC := ModuleCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat) ↦
                        (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddCommGrpCat.carrier))
                    (M := ℤ) (↑n₁ : ℤ))
                  (op (Functor.obj (singleFunctor C 0) S.X₃))) :
              Type (max u v))) :
        (a : ShiftedHom ((singleFunctor C 0).obj S.X₁) ((singleFunctor C 0).obj Y) (↑n₀ : ℤ)) →
          ShiftedHom ((singleFunctor C 0).obj S.X₃) ((singleFunctor C 0).obj Y) (↑n₁ : ℤ))
      x.hom :
    ShiftedHom (C := DerivedCategory C) (M := ℤ) (Functor.obj (singleFunctor C 0) S.X₃)
      (Functor.obj (singleFunctor C 0) Y) (↑n₁ : ℤ))
  ((comp hS.extClass x h).hom :
    ShiftedHom (C := DerivedCategory C) (M := ℤ) (Functor.obj (singleFunctor C 0) S.X₃)
      (Functor.obj (singleFunctor C 0) Y) (↑n₁ : ℤ))
Tactic: dsimp