-- 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 : C
rw :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (Iso.hom (ρ_ (F.obj X)) ≫ F.map (Iso.inv (ρ_ X)))
rw₁ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (F.obj X ◁ ε F) (μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫ F.map (Iso.hom (ρ_ X))) ≫
      F.map (Iso.inv (ρ_ X)))
rw₂ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
      (F.obj X ◁ ε F) ((μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫ F.map (Iso.hom (ρ_ X))) ≫ F.map (Iso.inv (ρ_ X))))
rw₃ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
      (F.obj X ◁ ε F) (μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫ F.map (Iso.hom (ρ_ X)) ≫ F.map (Iso.inv (ρ_ X))))
rw₄ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
      (F.obj X ◁ ε F) (μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫ F.map (Iso.hom (ρ_ X) ≫ Iso.inv (ρ_ X))))
rw₅ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (F.obj X ◁ ε F ≫
      μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫
        F.map (𝟙 (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))))
rw₆ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (F.obj X ◁ ε F ≫
      μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫
        𝟙 (F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))))
rw₇ rw₈ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
⊢ Eq (α := F.obj X ⟶ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)))
      (Iso.inv (ρ_ (F.obj X))) (F.obj X ◁ ε F ≫ μ F X (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (F.map (Iso.inv (ρ_ X)))

Tactic:
rw [Iso.inv_comp_eq, right_unitality, Category.assoc, Category.assoc, ← F.map_comp,
    Iso.hom_inv_id, F.map_id, comp_id]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Functor.LaxMonoidal.right_unitality, CategoryTheory.Iso.inv_comp_eq, CategoryTheory.Iso.hom_inv_id, CategoryTheory.Category.comp_id, CategoryTheory.Functor.map_id, CategoryTheory.Functor.map_comp