-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : F.LaxMonoidal
X Y Z : C
rw :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) (F.obj Z) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z))
    (Iso.hom (α_ (F.obj X) (F.obj Y) (F.obj Z)) ≫
      F.obj X ◁ μ F Y Z ≫ μ F X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z) ≫ F.map (Iso.inv (α_ X Y Z)))
    (μ F X Y ▷ F.obj Z ≫ μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)
rw₁ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) (F.obj Z) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z))
    (μ F X Y ▷ F.obj Z ≫
      μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z ≫ F.map (Iso.hom (α_ X Y Z)) ≫ F.map (Iso.inv (α_ X Y Z)))
    (μ F X Y ▷ F.obj Z ≫ μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)
rw₂ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) (F.obj Z) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z))
    (μ F X Y ▷ F.obj Z ≫ μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z ≫ F.map (Iso.hom (α_ X Y Z) ≫ Iso.inv (α_ X Y Z)))
    (μ F X Y ▷ F.obj Z ≫ μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)
rw₃ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) (F.obj Z) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z))
    (μ F X Y ▷ F.obj Z ≫
      μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z ≫
        F.map (𝟙 (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)))
    (μ F X Y ▷ F.obj Z ≫ μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)
rw₄ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) (F.obj Z) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z))
    (μ F X Y ▷ F.obj Z ≫
      μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z ≫
        𝟙 (F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)))
    (μ F X Y ▷ F.obj Z ≫ μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)
rw₅ rw₆ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) (F.obj Z) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z))
    (μ F X Y ▷ F.obj Z ≫ μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)
    (μ F X Y ▷ F.obj Z ≫ μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj Y) (F.obj Z)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z))
    (F.obj X ◁ μ F Y Z ≫ μ F X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z) ≫ F.map (Iso.inv (α_ X Y Z)))
    (Iso.inv (α_ (F.obj X) (F.obj Y) (F.obj Z)) ≫ μ F X Y ▷ F.obj Z ≫ μ F (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z)

Tactic:
rw [Iso.eq_inv_comp, ← associativity_assoc, ← F.map_comp, Iso.hom_inv_id,
    F.map_id, comp_id]

Premises used: CategoryTheory.Functor.LaxMonoidal.associativity_assoc, CategoryTheory.Iso.hom_inv_id, CategoryTheory.Category.comp_id, CategoryTheory.Functor.map_id, CategoryTheory.Functor.map_comp, CategoryTheory.Iso.eq_inv_comp