-- Module: Mathlib.CategoryTheory.GradedObject.Monoidal
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
X₁ X₂ X₃ X₄ : GradedObject I C
inst_3 : HasTensor.{u, v_1, u_1} X₁ X₂
inst_4 : HasTensor.{u, v_1, u_1} X₂ X₃
inst_5 : HasTensor.{u, v_1, u_1} X₃ X₄
inst_6 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃
inst_7 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃)
inst_8 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_9 : HasTensor.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_10 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃) X₄
inst_11 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃)) X₄
inst_12 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₂ X₃) X₄)
inst_13 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄))
inst_14 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_15 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ X₂ X₃
inst_16 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ X₂ X₃
inst_17 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_18 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_19 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₂ X₃ X₄
inst_20 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₂ X₃ X₄
inst_21 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃ X₄
inst_22 : HasGoodTensorTensor₂₃.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃ X₄
inst_23 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_24 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_25 : HasTensor₄ObjExt.{u, v_1, u_1} X₁ X₂ X₃ X₄
j i₁ i₂ i₃ i₄ : I
h : i₁ + i₂ + i₃ + i₄ = j
rw :
  Iso.inv (α_ (X₁ i₁) (X₂ i₂) (MonoidalCategoryStruct.tensorObj.{v_1, u_1} (𝒞 := inst_1) (X₃ i₃) (X₄ i₄))) ≫
    Iso.inv (α_ (MonoidalCategoryStruct.tensorObj.{v_1, u_1} (𝒞 := inst_1) (X₁ i₁) (X₂ i₂)) (X₃ i₃) (X₄ i₄)) ≫
      ιTensorObj₃' X₁ X₂ X₃ i₁ i₂ i₃ (i₁ + i₂ + i₃) ⋯ ▷ X₄ i₄ ≫
        ιTensorObj (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃) X₄ (i₁ + i₂ + i₃) i₄ j ⋯
| CategoryStruct.comp (Y := tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) j)
    (ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h)
    (CategoryStruct.comp (Y :=
      tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₂ X₃) X₄) j)
      (tensorHom (𝟙 X₁) (Iso.inv (associator X₂ X₃ X₄)) j)
      (CategoryStruct.comp (Y :=
        tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃)) X₄ j)
        (Iso.inv (associator X₁ (tensorObj.{u, v_1, u_1} X₂ X₃) X₄) j)
        (tensorHom (Iso.inv (associator X₁ X₂ X₃)) (𝟙 X₄) j)))

Tactic:
rw [ιTensorObj₄_eq X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h _ rfl, assoc, ι_tensorHom_assoc]
    dsimp only [categoryOfGradedObjects_id, id_eq, eq_mpr_eq_cast, cast_eq]
    rw [id_tensorHom, ← MonoidalCategory.whiskerLeft_comp_assoc, ιTensorObj₃_associator_inv,
      ιTensorObj₃'_eq X₂ X₃ X₄ i₂ i₃ i₄ _ rfl _ rfl, MonoidalCategory.whiskerLeft_comp_assoc,
      MonoidalCategory.whiskerLeft_comp_assoc,
      ← ιTensorObj₃_eq_assoc X₁ (tensorObj X₂ X₃) X₄ i₁ (i₂ + i₃) i₄ j
        (by simp only [← add_assoc, h]) _ rfl, ιTensorObj₃_associator_inv_assoc,
      ιTensorObj₃'_eq_assoc X₁ (tensorObj X₂ X₃) X₄ i₁ (i₂ + i₃) i₄ j
        (by simp only [← add_assoc, h]) (i₁ + i₂ + i₃) (by rw [add_assoc]), ι_tensorHom]
    dsimp only [id_eq, eq_mpr_eq_cast, categoryOfGradedObjects_id]
    rw [tensorHom_id, whisker_assoc_symm_assoc, Iso.hom_inv_id_assoc,
      ← MonoidalCategory.comp_whiskerRight_assoc, ← MonoidalCategory.comp_whiskerRight_assoc,
      ← ιTensorObj₃_eq X₁ X₂ X₃ i₁ i₂ i₃ _ rfl _ rfl, ιTensorObj₃_associator_inv,
      MonoidalCategory.comp_whiskerRight_assoc, MonoidalCategory.pentagon_inv_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.id_tensorHom, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃_associator_inv, CategoryTheory.GradedObject.categoryOfGradedObjects_id, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃_associator_inv_assoc, CategoryTheory.GradedObject.Monoidal.tensorObj, CategoryTheory.MonoidalCategory.pentagon_inv_assoc, CategoryTheory.MonoidalCategory.comp_whiskerRight_assoc, CategoryTheory.MonoidalCategory.whisker_assoc_symm_assoc, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃_eq, HAdd.hAdd, CategoryTheory.MonoidalCategory.whiskerLeft_comp_assoc, cast_eq, CategoryTheory.Iso.hom_inv_id_assoc, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃_eq_assoc, eq_mpr_eq_cast, CategoryTheory.GradedObject.Monoidal.ιTensorObj₄_eq, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃'_eq, CategoryTheory.GradedObject.Monoidal.ι_tensorHom_assoc, CategoryTheory.GradedObject.Monoidal.ιTensorObj₃'_eq_assoc, CategoryTheory.GradedObject.Monoidal.ι_tensorHom, rfl, CategoryTheory.MonoidalCategory.tensorHom_id, id_eq