-- Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Mon
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
inst_2 : Category.{w, u_2} D
inst_3 : CartesianMonoidalCategory.{w, u_2} D
M N O X Y : C
inst_4 : MonObj M
inst_5 : MonObj N
inst_6 : MonObj O
F : Functor.{v, w, u_1, w + 1} Cᵒᵖ MonCat
α :
  (F ⋙
        forget MonCat (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC :=
          MonCat.carrier)).RepresentableBy
    X
a :
  HMul.hMul (α := (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))) : Type w)) (β :=
      (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))) : Type w))
      (((α.homEquiv' (CD := MonCat.carrier) (FD := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) :
              Equiv.{v + 1, w + 1} (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst)) ⟶ X)
                (ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
                  (F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))))) :
            (a : X ⊗ 𝟙_ C ⟶ X) → ToType (F.obj (op (X ⊗ 𝟙_ C))))
          (fst X (tensorUnit.{v, u_1} C (𝒞 := inst))) :
        ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
          (F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
              (X := F.obj (op (tensorUnit.{v, u_1} C (𝒞 := inst)))) (Y :=
              F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst)))))
              (F.map (Quiver.Hom.op (snd X (tensorUnit.{v, u_1} C (𝒞 := inst))))) :
            (a : (↑(F.obj (op (𝟙_ C))) : Type w)) → (↑(F.obj (op (X ⊗ 𝟙_ C))) : Type w))
          1 :
        (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))) : Type w)) =
    ((α.homEquiv' (CD := MonCat.carrier) (FD := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) :
          Equiv.{v + 1, w + 1} (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst)) ⟶ X)
            (ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
              (F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))))) :
        (a : X ⊗ 𝟙_ C ⟶ X) → ToType (F.obj (op (X ⊗ 𝟙_ C))))
      (Iso.hom (ρ_ X))
⊢ HMul.hMul (α := (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))) : Type w)) (β :=
      (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))) : Type w))
      (((α.homEquiv' (CD := MonCat.carrier) (FD := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) :
              Equiv.{v + 1, w + 1} (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst)) ⟶ X)
                (ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
                  (F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))))) :
            (a : X ⊗ 𝟙_ C ⟶ X) → ToType (F.obj (op (X ⊗ 𝟙_ C))))
          (X ◁
              ((Equiv.symm
                      (α.homEquiv' (CD := MonCat.carrier) (FD := fun (x1 x2 : MonCat.{w}) ↦
                        (↑x1 : Type w) →* (↑x2 : Type w))) :
                    Equiv.{w + 1, v + 1}
                      (ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
                        (F.obj (op (tensorUnit.{v, u_1} C (𝒞 := inst)))))
                      (tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ X)) :
                  (a : ToType (F.obj (op (𝟙_ C)))) → 𝟙_ C ⟶ X)
                1 ≫
            fst X X) :
        ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
          (F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))))
      (((α.homEquiv' (CD := MonCat.carrier) (FD := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) :
              Equiv.{v + 1, w + 1} (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst)) ⟶ X)
                (ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
                  (F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))))) :
            (a : X ⊗ 𝟙_ C ⟶ X) → ToType (F.obj (op (X ⊗ 𝟙_ C))))
          (X ◁
              ((Equiv.symm
                      (α.homEquiv' (CD := MonCat.carrier) (FD := fun (x1 x2 : MonCat.{w}) ↦
                        (↑x1 : Type w) →* (↑x2 : Type w))) :
                    Equiv.{w + 1, v + 1}
                      (ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
                        (F.obj (op (tensorUnit.{v, u_1} C (𝒞 := inst)))))
                      (tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ X)) :
                  (a : ToType (F.obj (op (𝟙_ C)))) → 𝟙_ C ⟶ X)
                1 ≫
            snd X X) :
        ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
          (F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst)))))) =
    ((α.homEquiv' (CD := MonCat.carrier) (FD := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) :
          Equiv.{v + 1, w + 1} (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst)) ⟶ X)
            (ToType (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
              (F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorUnit.{v, u_1} C (𝒞 := inst))))))) :
        (a : X ⊗ 𝟙_ C ⟶ X) → ToType (F.obj (op (X ⊗ 𝟙_ C))))
      (Iso.hom (ρ_ X))

Tactic:
simp only [whiskerLeft_fst, whiskerLeft_snd, α.homEquiv'_comp, Equiv.apply_symm_apply]

Premises used: Equiv.apply_symm_apply, CategoryTheory.CartesianMonoidalCategory.whiskerLeft_snd, CategoryTheory.Functor.RepresentableBy.homEquiv'_comp, CategoryTheory.CartesianMonoidalCategory.whiskerLeft_fst