-- Module: Mathlib.Algebra.Category.HopfAlgCat.Monoidal
Goal:
R : Type u
inst : CommRing R
x x_1 x_2 : HopfAlgCat R
x_3 : x.carrier
x_4 : x_1.carrier
x_5 : x_2.carrier
⊢ Eq (α :=
    (Functor.obj
        (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC := carrier)
          (BialgCat R) (FD := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD := BialgCat.carrier))
        (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier)
    ((((((LinearMap.compr₂ₛₗ (TensorProduct.mk R x.carrier x_1.carrier)
                        (LinearMap.compr₂ₛₗ (σ₃₄ := RingHom.id R)
                          (TensorProduct.mk R (tensorObj (𝒞 := category) x x_1).carrier x_2.carrier)
                          ((fun
                                (x_6 :
                                  (Functor.obj
                                        (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                          x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                          fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                          BialgCat.carrier))
                                        (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) x x_1)
                                          x_2)).carrier →ₐc[R]
                                    (Functor.obj
                                        (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                          x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                          fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                          BialgCat.carrier))
                                        (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier) ↦
                              ((↑x_6 :
                                  [inst_1 :
                                      FunLike
                                        (((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier →ₐc[R]
                                          ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
                                        ((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier
                                        ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier] →
                                    [SemilinearMapClass
                                          (((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier →ₐc[R]
                                            ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
                                          (RingHom.id R)
                                          ((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier
                                          ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier] →
                                      ((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier →ₗ[R]
                                        ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier) :
                                (Functor.obj
                                      (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                        x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                        fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                        BialgCat.carrier))
                                      (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) x x_1) x_2)).carrier →ₗ[R]
                                  (Functor.obj
                                      (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                        x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                        fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                        BialgCat.carrier))
                                      (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier))
                            (BialgCat.Hom.toBialgHom'
                              (CategoryTheory.Functor.map
                                (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                  x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                  fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD := BialgCat.carrier))
                                (Iso.hom (α_ x x_1 x_2)))))) :
                      (a : x.carrier) →
                        x_1.carrier →ₗ[R]
                          x_2.carrier →ₗ[R] ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
                    x_3 :
                  x_1.carrier →ₗ[R]
                    x_2.carrier →ₗ[R]
                      (Functor.obj
                          (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                            (CC := carrier) (BialgCat R) (FD := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                            (CD := BialgCat.carrier))
                          (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier) :
                (a : x_1.carrier) →
                  x_2.carrier →ₗ[R] ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
              x_4 :
            x_2.carrier →ₗ[R]
              (Functor.obj
                  (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                    carrier) (BialgCat R) (FD := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                    BialgCat.carrier))
                  (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier) :
          (a : x_2.carrier) → ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
        x_5 :
      (Functor.obj
          (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC := carrier)
            (BialgCat R) (FD := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD := BialgCat.carrier))
          (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier)
    ((((((LinearMap.compr₂ₛₗ (TensorProduct.mk R x.carrier x_1.carrier)
                        (LinearMap.compr₂ₛₗ (σ₃₄ := RingHom.id R)
                          (TensorProduct.mk R (tensorObj (𝒞 := category) x x_1).carrier x_2.carrier)
                          ((fun
                                (x_6 :
                                  (Functor.obj
                                        (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                          x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                          fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                          BialgCat.carrier))
                                        (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) x x_1)
                                          x_2)).carrier →ₐc[R]
                                    (Functor.obj
                                        (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                          x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                          fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                          BialgCat.carrier))
                                        (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier) ↦
                              ((↑x_6 :
                                  [inst_1 :
                                      FunLike
                                        (((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier →ₐc[R]
                                          ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
                                        ((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier
                                        ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier] →
                                    [SemilinearMapClass
                                          (((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier →ₐc[R]
                                            ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
                                          (RingHom.id R)
                                          ((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier
                                          ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier] →
                                      ((forget₂ (HopfAlgCat R) (BialgCat R)).obj ((x ⊗ x_1) ⊗ x_2)).carrier →ₗ[R]
                                        ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier) :
                                (Functor.obj
                                      (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                        x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                        fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                        BialgCat.carrier))
                                      (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) x x_1) x_2)).carrier →ₗ[R]
                                  (Functor.obj
                                      (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                        x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                        fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                        BialgCat.carrier))
                                      (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier))
                            ((Iso.symm
                                      (Iso.refl
                                        (tensorObj (𝒞 := BialgCat.category)
                                          (Functor.obj
                                            (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                              x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                              fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                              BialgCat.carrier))
                                            (tensorObj (𝒞 := category) x x_1))
                                          (Functor.obj
                                            (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                              x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                              fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                              BialgCat.carrier))
                                            x_2))) ≪≫
                                    (Iso.symm
                                        (Iso.refl
                                          (tensorObj (𝒞 := BialgCat.category)
                                            (Functor.obj
                                              (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                                x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                                fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                                BialgCat.carrier))
                                              x)
                                            (Functor.obj
                                              (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                                x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                                fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                                BialgCat.carrier))
                                              x_1))) ⊗ᵢ
                                      Iso.refl
                                        (Functor.obj
                                          (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                            x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                            fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                            BialgCat.carrier))
                                          x_2))) ≪≫
                                  α_
                                      (Functor.obj
                                        (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                          x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                          fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                          BialgCat.carrier))
                                        x)
                                      (Functor.obj
                                        (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                          x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                          fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                          BialgCat.carrier))
                                        x_1)
                                      (Functor.obj
                                        (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                          x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                          fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                          BialgCat.carrier))
                                        x_2) ≪≫
                                    (Iso.refl
                                          (Functor.obj
                                            (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                              x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                              fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                              BialgCat.carrier))
                                            x) ⊗ᵢ
                                        Iso.refl
                                          (tensorObj (𝒞 := BialgCat.category)
                                            (Functor.obj
                                              (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                                x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                                fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                                BialgCat.carrier))
                                              x_1)
                                            (Functor.obj
                                              (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                                x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                                fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                                BialgCat.carrier))
                                              x_2))) ≪≫
                                      Iso.refl
                                        (tensorObj (𝒞 := BialgCat.category)
                                          (Functor.obj
                                            (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                              x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                              fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                              BialgCat.carrier))
                                            x)
                                          (Functor.obj
                                            (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦
                                              x1.carrier →ₐc[R] x2.carrier) (CC := carrier) (BialgCat R) (FD :=
                                              fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                                              BialgCat.carrier))
                                            (tensorObj (𝒞 := category) x_1 x_2)))).hom.toBialgHom')) :
                      (a : x.carrier) →
                        x_1.carrier →ₗ[R]
                          x_2.carrier →ₗ[R] ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
                    x_3 :
                  x_1.carrier →ₗ[R]
                    x_2.carrier →ₗ[R]
                      (Functor.obj
                          (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                            (CC := carrier) (BialgCat R) (FD := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                            (CD := BialgCat.carrier))
                          (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier) :
                (a : x_1.carrier) →
                  x_2.carrier →ₗ[R] ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
              x_4 :
            x_2.carrier →ₗ[R]
              (Functor.obj
                  (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                    carrier) (BialgCat R) (FD := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD :=
                    BialgCat.carrier))
                  (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier) :
          (a : x_2.carrier) → ((forget₂ (HopfAlgCat R) (BialgCat R)).obj (x ⊗ x_1 ⊗ x_2)).carrier)
        x_5 :
      (Functor.obj
          (forget₂ (HopfAlgCat R) (FC := fun (x1 x2 : HopfAlgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC := carrier)
            (BialgCat R) (FD := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CD := BialgCat.carrier))
          (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))).carrier)

Tactic:
rfl