-- 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 Z
inst_6 : HasTensor.{u_1, v_1, u_2} Z X
inst_7 : HasTensor.{u_1, v_1, u_2} Z Y
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} Z (tensorObj.{u_1, v_1, u_2} X Y)
inst_12 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y
inst_13 : HasTensor.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} Z Y)
inst_14 : HasTensor.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y
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} Z X Y
inst_18 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} Z X Y
inst_19 : HasGoodTensor₁₂Tensor.{u_1, v_1, u_2} X Z Y
inst_20 : HasGoodTensorTensor₂₃.{u_1, v_1, u_2} X Z Y
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) (X i₁)
        (MonoidalCategoryStruct.tensorObj.{v_1, u_2} (𝒞 := inst_1) (Y i₂) (Z i₃)) ⟶
      tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y k)
    (MonoidalCategoryStruct.whiskerLeft (X i₁) (Iso.hom (β_ (Y i₂) (Z i₃))) ≫
      Iso.inv (MonoidalCategoryStruct.associator (X i₁) (Z i₃) (Y i₂)) ≫
        MonoidalCategoryStruct.whiskerRight (Iso.hom (β_ (X i₁) (Z i₃))) (Y i₂) ≫ ιTensorObj₃' Z X Y i₃ i₁ i₂ k ⋯)
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} 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} Z Y) k)
        (whiskerLeft X
          (fun (k : I) ↦
            tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (Y i) (Z j)) ≫ ιTensorObj Z Y j i k ⋯)
          k)
        (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y k)
          (Iso.inv (associator X Z Y) k)
          (whiskerRight
            (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 ⋯)
            Y k))))
⊢ Eq.{v_1 + 1} (α :=
    MonoidalCategoryStruct.tensorObj.{v_1, u_2} (𝒞 := inst_1) (X i₁)
        (MonoidalCategoryStruct.tensorObj.{v_1, u_2} (𝒞 := inst_1) (Y i₂) (Z i₃)) ⟶
      tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} Z X) Y k)
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} 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} X Y) Z k)
        (Iso.inv (associator X Y Z) k)
        (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y) k)
          (tensorObjDesc fun (i j : I) (hij : i + j = k) ↦
            Iso.hom (β_ (tensorObj.{u_1, v_1, u_2} X Y i) (Z j)) ≫ ιTensorObj Z (tensorObj.{u_1, v_1, u_2} X Y) j i k ⋯)
          (Iso.inv (associator Z X Y) k))))
    (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} X (tensorObj.{u_1, v_1, u_2} 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} Z Y) k)
        (whiskerLeft X
          (fun (k : I) ↦
            tensorObjDesc fun (i j : I) (hij : i + j = k) ↦ Iso.hom (β_ (Y i) (Z j)) ≫ ιTensorObj Z Y j i k ⋯)
          k)
        (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y k)
          (Iso.inv (associator X Z Y) k)
          (whiskerRight
            (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 ⋯)
            Y k))))

Tactic:
conv_lhs => rw [ιTensorObj₃_associator_inv_assoc, ιTensorObj₃'_eq X Y Z i₁ i₂ i₃ k h _ rfl, assoc,
    ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.tensorHom_id,
    BraidedCategory.braiding_naturality_assoc,
    BraidedCategory.braiding_tensor_left_hom, assoc, assoc, assoc, assoc, Iso.inv_hom_id_assoc,
    MonoidalCategory.id_tensorHom,
    ← ιTensorObj₃_eq_assoc Z X Y i₃ i₁ i₂ k (by rw [add_assoc, add_comm i₃, h]) _ rfl,
    ιTensorObj₃_associator_inv, Iso.hom_inv_id_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.id_tensorHom, CategoryTheory.BraidedCategory.braiding_tensor_left_hom, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃_associator_inv, CategoryTheory.GradedObject.Monoidal.ι_tensorObjDesc_assoc, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃_associator_inv_assoc, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.BraidedCategory.braiding_naturality_assoc, CategoryTheory.Iso.hom_inv_id_assoc, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃_eq_assoc, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃'_eq, rfl, CategoryTheory.MonoidalCategory.tensorHom_id