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

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

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.eq_comp_inv, CategoryTheory.Iso.hom_inv_id, CategoryTheory.Functor.map_id, CategoryTheory.Functor.map_comp, CategoryTheory.Functor.OplaxMonoidal.right_unitality, CategoryTheory.Category.id_comp