-- Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Grp
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : CartesianMonoidalCategory.{v, u} C
M G H X Y : C
inst_2 : MonObj M
inst_3 : GrpObj G
inst_4 : GrpObj H
F : Functor.{v, w, u, w + 1} Cᵒᵖ GrpCat
α :
  (F ⋙
        forget GrpCat (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC :=
          GrpCat.carrier)).RepresentableBy
    X
change :
  Eq.{v + 1} (α := X ⟶ X)
    (CategoryStruct.comp (Y := tensorObj.{v, u} (𝒞 := inst) X X)
      (lift (𝟙 X)
        (((Equiv.symm
                (α.homEquiv' (CD := GrpCat.carrier) (FD := fun (x1 x2 : GrpCat.{w}) ↦
                  (↑x1 : Type w) →* (↑x2 : Type w))) :
              Equiv.{w + 1, v + 1}
                (ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
                  (F.obj (op X)))
                (X ⟶ X)) :
            (a : ToType (F.obj (op X))) → X ⟶ X)
          (((α.homEquiv' (CD := GrpCat.carrier) (FD := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) :
                  Equiv.{v + 1, w + 1} (X ⟶ X)
                    (ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
                      (F.obj (op X)))) :
                (a : X ⟶ X) → ToType (F.obj (op X)))
              (𝟙 X))⁻¹))
      (((Equiv.symm
              (α.homEquiv' (CD := GrpCat.carrier) (FD := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w))) :
            Equiv.{w + 1, v + 1}
              (ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
                (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
              (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)) :
          (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
        (HMul.hMul (α :=
          ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
            (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
          (β :=
          ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
            (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
          (((α.homEquiv' (CD := GrpCat.carrier) (FD := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) :
                  Equiv.{v + 1, w + 1} (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
                    (ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
                      (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))) :
                (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
              (fst X X) :
            ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
              (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
          (((α.homEquiv' (CD := GrpCat.carrier) (FD := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) :
                  Equiv.{v + 1, w + 1} (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
                    (ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
                      (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))) :
                (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
              (snd X X) :
            ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
              (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X)))))))
    (CategoryStruct.comp (Y := tensorUnit.{v, u} C (𝒞 := inst)) (toUnit X)
      (((Equiv.symm
              (α.homEquiv' (CD := GrpCat.carrier) (FD := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w))) :
            Equiv.{w + 1, v + 1}
              (ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
                (F.obj (op (tensorUnit.{v, u} C (𝒞 := inst)))))
              (tensorUnit.{v, u} C (𝒞 := inst) ⟶ X)) :
          (a : ToType (F.obj (op (𝟙_ C)))) → 𝟙_ C ⟶ X)
        1))
⊢ Eq.{v + 1} (α := X ⟶ X)
    (CategoryStruct.comp (Y := tensorObj.{v, u} (𝒞 := inst) X X)
      (lift (𝟙 X)
        (((Equiv.symm
                (α.homEquiv' (CD := GrpCat.carrier) (FD := fun (x1 x2 : GrpCat.{w}) ↦
                  (↑x1 : Type w) →* (↑x2 : Type w))) :
              Equiv.{w + 1, v + 1}
                (ToType (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := GrpCat.carrier)
                  (F.obj (op X)))
                (X ⟶ X)) :
            (a : ToType (F.obj (op X))) → X ⟶ X)
          (((α.homEquiv :
                  Equiv.{v + 1, w + 1} (X ⟶ X)
                    ((F ⋙
                          forget GrpCat (FC := fun (x1 x2 : GrpCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC :=
                            GrpCat.carrier)).obj
                      (op X))) :
                (a : X ⟶ X) → (F ⋙ forget GrpCat).obj (op X))
              (𝟙 X))⁻¹))
      μ)
    (CategoryStruct.comp (Y := tensorUnit.{v, u} C (𝒞 := inst)) (toUnit X) η)

Tactic:
change lift (𝟙 X) (α.homEquiv'.symm (α.homEquiv' (𝟙 X))⁻¹) ≫
      α.homEquiv'.symm (α.homEquiv' (fst X X) * α.homEquiv' (snd X X)) =
        toUnit X ≫ α.homEquiv'.symm 1

Premises used: HMul.hMul, CategoryTheory.Functor.RepresentableBy.homEquiv', CategoryTheory.CategoryStruct.id, CategoryTheory.SemiCartesianMonoidalCategory.snd, CategoryTheory.CartesianMonoidalCategory.lift, Inv.inv, Equiv.symm, OfNat.ofNat, DFunLike.coe, CategoryTheory.SemiCartesianMonoidalCategory.toUnit, CategoryTheory.SemiCartesianMonoidalCategory.fst