-- 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)
    (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))))
⊢ 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)
        (Iso.inv (associator X Y Z))
        (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} Z (tensorObj.{u_1, v_1, u_2} X Y))
          (Iso.hom (braiding (tensorObj.{u_1, v_1, u_2} X Y) Z)) (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))
        (whiskerLeft X (Iso.hom (braiding Y Z)))
        (CategoryStruct.comp (Y := tensorObj.{u_1, v_1, u_2} (tensorObj.{u_1, v_1, u_2} X Z) Y)
          (Iso.inv (associator X Z Y)) (whiskerRight (Iso.hom (braiding X Z)) Y))
        k))

Tactic:
dsimp [braiding]

Premises used: CategoryTheory.GradedObject.Monoidal.braiding