-- Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasFiniteCoproducts.{v_1, u_1} C
P : Karoubi (SimplicialObject.{v_1, u_1} C)
n : ℕ
eq₁ :
  Eq (α :=
    HomologicalComplex.X (Karoubi.X (Functor.obj (𝟭 (Karoubi (ChainComplex.{v_1, u_1, 0} C ℕ))) (Functor.obj N₂ P))) n ⟶
      HomologicalComplex.X
        (Karoubi.X (Functor.obj (C := Karoubi (ChainComplex.{v_1, u_1, 0} C ℕ)) (Γ₂ ⋙ N₂) (Functor.obj N₂ P))) n)
    (HomologicalComplex.Hom.f (Karoubi.Hom.f (NatTrans.app (Iso.inv N₂Γ₂) (Functor.obj N₂ P))) n)
    (HomologicalComplex.Hom.f PInfty n ≫
      P.p.app (op ⦋n⦌) ≫
        Cofan.inj (Splitting.cofan (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))) (op ⦋n⦌))
          (Splitting.IndexSet.id (op ⦋n⦌)))
rw :
  Eq (α :=
    Splitting.N (X := Γ₀.obj (Karoubi.X (Functor.obj N₂ P))) (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))) n ⟶ P.X _⦋n⦌)
    (CategoryStruct.comp (Y :=
        P.X.obj
          (Sigma.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α })
            (Splitting.IndexSet.id (op ⦋n⦌))))
        (HomologicalComplex.Hom.f PInfty
            (unop
                (Sigma.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α })
                  (Splitting.IndexSet.id (op ⦋n⦌)))).len :
          HomologicalComplex.X K[P.X]
              (unop
                  (Sigma.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α })
                    (Splitting.IndexSet.id (op ⦋n⦌)))).len ⟶
            HomologicalComplex.X K[P.X]
              (unop
                  (Sigma.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α })
                    (Splitting.IndexSet.id (op ⦋n⦌)))).len)
        (P.p.app
            (Sigma.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α })
              (Splitting.IndexSet.id (op ⦋n⦌))) ≫
          CategoryStruct.comp (Y :=
            Γ₀.Obj.obj₂
              (Karoubi.X (Functor.obj N₂ { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) }))
              (op ⦋n⦌))
            (Cofan.inj
                (Splitting.cofan
                  (Γ₀.splitting
                    (Karoubi.X
                      (Functor.obj N₂ { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) })))
                  (op ⦋n⦌))
                (Splitting.IndexSet.id (op ⦋n⦌)) :
              Splitting.summand
                  (Splitting.N (X :=
                    Γ₀.obj
                      (Karoubi.X
                        (Functor.obj N₂ { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) })))
                    (Γ₀.splitting
                      (Karoubi.X
                        (Functor.obj N₂
                          { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) }))))
                  (op ⦋n⦌) (Splitting.IndexSet.id (op ⦋n⦌)) ⟶
                Cocone.pt (J := Discrete (Splitting.IndexSet (op ⦋n⦌))) (F :=
                  Discrete.functor
                    (Splitting.summand
                      (Splitting.N (X :=
                        Γ₀.obj
                          (Karoubi.X
                            (Functor.obj N₂
                              { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) })))
                        (Γ₀.splitting
                          (Karoubi.X
                            (Functor.obj N₂
                              { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) }))))
                      (op ⦋n⦌)))
                  (Splitting.cofan
                    (Γ₀.splitting
                      (Karoubi.X
                        (Functor.obj N₂ { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) })))
                    (op ⦋n⦌)))
            (CategoryStruct.comp (Y := Γ₀.Obj.obj₂ K[P.X] (op ⦋n⦌))
                (NatTrans.app
                    (Karoubi.Hom.f (CategoryTheory.Functor.map Γ₂ (NatTrans.app (Iso.hom toKaroubiCompN₂IsoN₁) P.X)))
                    (op ⦋n⦌) :
                  (Functor.obj (C := Karoubi (ChainComplex.{v_1, u_1, 0} C ℕ)) Γ₂
                          (Functor.obj N₂
                            { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) })).X _⦋n⦌ ⟶
                    (Functor.obj (C := Karoubi (ChainComplex.{v_1, u_1, 0} C ℕ)) Γ₂ (Functor.obj N₁ P.X)).X _⦋n⦌)
                ((Splitting.desc (Γ₀.splitting K[P.X]) (op ⦋n⦌) fun (A : Splitting.IndexSet (op ⦋n⦌)) ↦
                    HomologicalComplex.Hom.f PInfty
                        (unop (A.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α }))).len ≫
                      P.X.map A.e.op) ≫
                  P.p.app (op ⦋n⦌)) :
              Γ₀.Obj.obj₂
                  (Karoubi.X
                    (Functor.obj N₂ { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) }))
                  (op ⦋n⦌) ⟶
                P.X _⦋n⦌)) :
      Splitting.summand
          (Splitting.N (X := Γ₀.obj (Karoubi.X (Functor.obj N₂ P))) (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))))
          (op ⦋n⦌) (Splitting.IndexSet.id (op ⦋n⦌)) ⟶
        P.X _⦋n⦌)
    (CategoryStruct.comp (Y := P.X _⦋n⦌)
        (HomologicalComplex.Hom.f PInfty n : HomologicalComplex.X K[P.X] n ⟶ HomologicalComplex.X K[P.X] n)
        (P.p.app (op ⦋n⦌)) :
      P.X _⦋n⦌ ⟶ P.X _⦋n⦌)
⊢ Eq (α :=
    Splitting.N (X := Γ₀.obj (Karoubi.X (Functor.obj N₂ P))) (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))) n ⟶ P.X _⦋n⦌)
    (CategoryStruct.comp (Y := Γ₀.Obj.obj₂ (Karoubi.X (Functor.obj N₂ P)) (op ⦋n⦌))
        (Cofan.inj (Splitting.cofan (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))) (op ⦋n⦌))
            (Splitting.IndexSet.id (op ⦋n⦌)) :
          Splitting.summand
              (Splitting.N (X := Γ₀.obj (Karoubi.X (Functor.obj N₂ P))) (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))))
              (op ⦋n⦌) (Splitting.IndexSet.id (op ⦋n⦌)) ⟶
            Cocone.pt (J := Discrete (Splitting.IndexSet (op ⦋n⦌))) (F :=
              Discrete.functor
                (Splitting.summand
                  (Splitting.N (X := Γ₀.obj (Karoubi.X (Functor.obj N₂ P)))
                    (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))))
                  (op ⦋n⦌)))
              (Splitting.cofan (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))) (op ⦋n⦌)))
        (CategoryStruct.comp (Y :=
            Γ₀.Obj.obj₂
              (Karoubi.X (Functor.obj N₂ { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) }))
              (op ⦋n⦌))
            (Splitting.desc (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))) (op ⦋n⦌)
                fun (A : Splitting.IndexSet (op ⦋n⦌)) ↦
                CategoryStruct.comp (Y :=
                    P.X.obj (A.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α })))
                    (HomologicalComplex.Hom.f PInfty
                        (unop (A.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α }))).len :
                      HomologicalComplex.X K[P.X]
                          (unop (A.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α }))).len ⟶
                        HomologicalComplex.X K[P.X]
                          (unop (A.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α }))).len)
                    (P.p.app (A.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α }))) ≫
                  Cofan.inj
                    (Splitting.cofan
                      (Γ₀.splitting
                        (Karoubi.X
                          (Functor.obj N₂
                            { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) })))
                      (op ⦋n⦌))
                    A :
              Γ₀.obj (Karoubi.X (Functor.obj N₂ P)) _⦋n⦌ ⟶
                Γ₀.Obj.obj₂
                  (Karoubi.X
                    (Functor.obj N₂ { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) }))
                  (op ⦋n⦌))
            (CategoryStruct.comp (Y := Γ₀.Obj.obj₂ K[P.X] (op ⦋n⦌))
                  (NatTrans.app (Karoubi.Hom.f (NatTrans.app (Iso.hom Γ₂N₂ToKaroubiIso) P.X)) (op ⦋n⦌) :
                    (Functor.obj (C := Karoubi (ChainComplex.{v_1, u_1, 0} C ℕ)) Γ₂
                            (Functor.obj N₂
                              { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) })).X _⦋n⦌ ⟶
                      (Functor.obj (C := Karoubi (ChainComplex.{v_1, u_1, 0} C ℕ)) Γ₂ (Functor.obj N₁ P.X)).X _⦋n⦌)
                  (Splitting.desc (Γ₀.splitting K[P.X]) (op ⦋n⦌) fun (A : Splitting.IndexSet (op ⦋n⦌)) ↦
                    HomologicalComplex.Hom.f PInfty
                        (unop (A.fst (β := fun (Δ' : SimplexCategoryᵒᵖ) ↦ { α : ⦋n⦌ ⟶ unop Δ' // Epi α }))).len ≫
                      P.X.map A.e.op) ≫
                P.p.app (op ⦋n⦌) :
              Γ₀.Obj.obj₂
                  (Karoubi.X
                    (Functor.obj N₂ { X := P.X, p := 𝟙 P.X, idem := ⋯ : Karoubi (SimplicialObject.{v_1, u_1} C) }))
                  (op ⦋n⦌) ⟶
                P.X _⦋n⦌) :
          Γ₀.Obj.obj₂ (Karoubi.X (Functor.obj N₂ P)) (op ⦋n⦌) ⟶ P.X _⦋n⦌) :
      Splitting.N (X := Γ₀.obj (Karoubi.X (Functor.obj N₂ P))) (Γ₀.splitting (Karoubi.X (Functor.obj N₂ P))) n ⟶
        P.X _⦋n⦌)
    (CategoryStruct.comp (Y := P.X _⦋n⦌)
        (HomologicalComplex.Hom.f PInfty n : HomologicalComplex.X K[P.X] n ⟶ HomologicalComplex.X K[P.X] n)
        (P.p.app (op ⦋n⦌)) :
      P.X _⦋n⦌ ⟶ P.X _⦋n⦌)

Tactic:
rw [Γ₂N₂ToKaroubiIso_hom_app, assoc, Splitting.ι_desc_assoc, assoc, assoc]

Premises used: CategoryTheory.Category.assoc, AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso_hom_app, CategoryTheory.SimplicialObject.Splitting.ι_desc_assoc