-- 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
E : Type u₃
inst_4 : Category.{v₃, u₃} E
inst_5 : MonoidalCategory.{v₃, u₃} E
C' : Type u₁'
inst_6 : Category.{v₁', u₁'} C'
F : Functor.{v₁, v₂, u₁, u₂} C D
G : Functor.{v₂, v₁, u₂, u₁} D C
adj : F ⊣ G
inst_7 : G.LaxMonoidal
x : C
rw :
  Eq.{v₁ + 1} (α := x ⟶ G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)))
    (((adj.homEquiv x (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)) :
            Equiv.{v₂ + 1, v₁ + 1}
              (F.obj x ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x))
              (x ⟶ G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)))) :
          (a : F.obj x ⟶ 𝟙_ D ⊗ F.obj x) → x ⟶ G.obj (𝟙_ D ⊗ F.obj x))
        (Iso.inv (λ_ (F.obj x))) :
      x ⟶ G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x) (Iso.inv (λ_ x))
        ((adj.unit.app (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫
              G.map
                (((Equiv.symm (adj.homEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2))) :
                      Equiv.{v₁ + 1, v₂ + 1}
                        (tensorUnit.{v₁, u₁} C (𝒞 := inst) ⟶ G.obj (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)))
                        (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2))) :
                    (a : 𝟙_ C ⟶ G.obj (𝟙_ D)) → F.obj (𝟙_ C) ⟶ 𝟙_ D)
                  (ε G)) ⊗ₘ
            adj.unit.app x ≫ 𝟙 (G.obj (F.obj x))) ≫
          μ G (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)) :
      x ⟶ G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)))
⊢ Eq.{v₁ + 1} (α := x ⟶ G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)))
    (((adj.homEquiv x (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)) :
            Equiv.{v₂ + 1, v₁ + 1}
              (F.obj x ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x))
              (x ⟶ G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)))) :
          (a : F.obj x ⟶ 𝟙_ D ⊗ F.obj x) → x ⟶ G.obj (𝟙_ D ⊗ F.obj x))
        (Iso.inv (λ_ (F.obj x))) :
      x ⟶ G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)))
    (((adj.homEquiv x (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)) :
            Equiv.{v₂ + 1, v₁ + 1}
              (F.obj x ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x))
              (x ⟶ G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)))) :
          (a : F.obj x ⟶ 𝟙_ D ⊗ F.obj x) → x ⟶ G.obj (𝟙_ D ⊗ F.obj x))
        (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
          (F.map (Iso.inv (λ_ x)))
          (CategoryStruct.comp (Y :=
            tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (F.obj x))
            (((Equiv.symm
                    (adj.homEquiv (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x)
                      (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (F.obj x))) :
                  Equiv.{v₁ + 1, v₂ + 1}
                    (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x ⟶
                      G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (F.obj x)))
                    (F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x) ⟶
                      tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (F.obj x))) :
                (a : 𝟙_ C ⊗ x ⟶ G.obj (F.obj (𝟙_ C) ⊗ F.obj x)) → F.obj (𝟙_ C ⊗ x) ⟶ F.obj (𝟙_ C) ⊗ F.obj x)
              ((adj.unit.app (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⊗ₘ adj.unit.app x) ≫
                μ G (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (F.obj x)))
            (((Equiv.symm (adj.homEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2))) :
                    Equiv.{v₁ + 1, v₂ + 1}
                      (tensorUnit.{v₁, u₁} C (𝒞 := inst) ⟶ G.obj (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)))
                      (F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2))) :
                  (a : 𝟙_ C ⟶ G.obj (𝟙_ D)) → F.obj (𝟙_ C) ⟶ 𝟙_ D)
                (ε G) ▷
              F.obj x))) :
      x ⟶ G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj x)))

Tactic:
rw [Adjunction.homEquiv_naturality_left, Adjunction.homEquiv_naturality_right,
      Equiv.apply_symm_apply, assoc, ← μ_natural_left, ← tensorHom_id,
      tensorHom_comp_tensorHom_assoc]

Premises used: CategoryTheory.Category.assoc, Equiv.apply_symm_apply, CategoryTheory.MonoidalCategory.tensorHom_comp_tensorHom_assoc, CategoryTheory.Functor.LaxMonoidal.μ_natural_left, CategoryTheory.Adjunction.homEquiv_naturality_right, CategoryTheory.Adjunction.homEquiv_naturality_left, CategoryTheory.MonoidalCategory.tensorHom_id