-- Module: Mathlib.CategoryTheory.GradedObject.Braiding
Goal:
I : Type u_1
inst : AddCommMonoid I
C : Type u_2
inst_1 : Category.{v_1, u_2} C
inst_2 : MonoidalCategory.{v_1, u_2} C
X Y Z : GradedObject I C
inst_3 : BraidedCategory.{v_1, u_2} C
inst_4 : HasTensor.{u_1, v_1, u_2} X Y
inst_5 : HasTensor.{u_1, v_1, u_2} Y X
inst_6 : HasTensor.{u_1, v_1, u_2} Y Z
inst_7 : HasTensor.{u_1, v_1, u_2} Z X
inst_8 : HasTensor.{u_1, v_1, u_2} X Z
inst_9 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z
inst_10 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z)
inst_11 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y Z) X
inst_12 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} Z X)
inst_13 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y X) Z
inst_14 : HasTensor.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} X Z)
inst_15 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Y Z
inst_16 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Y Z
inst_17 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y Z X
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y Z X
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} Y X Z
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Y X Z
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
h₁ :
  Eq.{v_1 + 1} (α :=
    MonoidalCategoryStruct.tensorObj.{v_1, u_2} (𝒞 := inst_1)
        (MonoidalCategoryStruct.tensorObj.{v_1, u_2} (𝒞 := inst_1) (X i₁) (Y i₂)) (Z i₃) ⟶
      tensorObj.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} Z X) k)
    (MonoidalCategoryStruct.whiskerRight (Iso.hom (β_ (X i₁) (Y i₂))) (Z i₃) ≫
      Iso.hom (MonoidalCategoryStruct.associator (Y i₂) (X i₁) (Z i₃)) ≫
        MonoidalCategoryStruct.whiskerLeft (Y i₂) (Iso.hom (β_ (X i₁) (Z i₃))) ≫ ιTensorObj₃ Y Z X i₂ i₃ i₁ k ⋯)
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z k)
      (ιTensorObj₃' X Y Z i₁ i₂ i₃ k h)
      (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y X) Z k)
        (whiskerRight
          (fun (k : I) ↦
            tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (X i) (Y j)) ≫ ιTensorObj Y X j i k ⋯)
          Z k)
        (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} X Z) k)
          (Iso.hom (associator Y X Z) k)
          (whiskerLeft Y
            (fun (k : I) ↦
              tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (X i) (Z j)) ≫ ιTensorObj Z X j i k ⋯)
            k))))
⊢ Eq.{v_1 + 1} (α :=
    MonoidalCategoryStruct.tensorObj.{v_1, u_2} (𝒞 := inst_1)
        (MonoidalCategoryStruct.tensorObj.{v_1, u_2} (𝒞 := inst_1) (X i₁) (Y i₂)) (Z i₃) ⟶
      tensorObj.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} Z X) k)
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z k)
      (ιTensorObj₃' X Y Z i₁ i₂ i₃ k h)
      (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Y Z) k)
        (Iso.hom (associator X Y Z) k)
        (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y Z) X k)
          (tensorObjDesc fun (i j : I) (hij : i + j = k) ↦
            Iso.hom (β_ (X i) (tensorObj.{u_1, v_1, u_2} Y Z j)) ≫ ιTensorObj (tensorObj.{u_1, v_1, u_2} Y Z) X j i k ⋯)
          (Iso.hom (associator Y Z X) k))))
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Y) Z k)
      (ιTensorObj₃' X Y Z i₁ i₂ i₃ k h)
      (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Y X) Z k)
        (whiskerRight
          (fun (k : I) ↦
            tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (X i) (Y j)) ≫ ιTensorObj Y X j i k ⋯)
          Z k)
        (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Y (tensorObj.{u_1, v_1, u_2} X Z) k)
          (Iso.hom (associator Y X Z) k)
          (whiskerLeft Y
            (fun (k : I) ↦
              tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (X i) (Z j)) ≫ ιTensorObj Z X j i k ⋯)
            k))))

Tactic:
conv_lhs => rw [ιTensorObj₃'_associator_hom_assoc, ιTensorObj₃_eq X Y Z i₁ i₂ i₃ k h _ rfl,
    assoc, ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.id_tensorHom,
    BraidedCategory.braiding_naturality_assoc,
    BraidedCategory.braiding_tensor_right_hom, assoc, assoc, assoc, assoc, Iso.hom_inv_id_assoc,
    MonoidalCategory.tensorHom_id,
    ← ιTensorObj₃'_eq_assoc Y Z X i₂ i₃ i₁ k (by rw [add_comm _ i₁, ← add_assoc, h]) _ rfl,
    ιTensorObj₃'_associator_hom, Iso.inv_hom_id_assoc]

Premises used: CategoryTheory.BraidedCategory.braiding_tensor_right_hom, CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.id_tensorHom, CategoryTheory.GradedObject.Monoidal.ι_tensorObjDesc_assoc, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.BraidedCategory.braiding_naturality_assoc, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃'_associator_hom, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃_eq, CategoryTheory.Iso.hom_inv_id_assoc, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃'_eq_assoc, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃'_associator_hom_assoc, rfl, CategoryTheory.MonoidalCategory.tensorHom_id