-- Lean 4 proof step from Mathlib.Algebra.Category.BialgCat.Monoidal (goal)
Goal:
R : Type u
inst : CommRing R
x x_1 x_2 : BialgCat R
⊢ Eq (α :=
    x_2.carrier →ₐ[R]
      (↑(Functor.obj
            (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC := carrier)
              (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
            (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))) :
        Type u))
    (AlgHom.comp
      (AlgHom.restrictScalars R (S := R)
        (AlgCat.Hom.hom
          (CategoryTheory.Functor.map
            (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC := carrier)
              (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
            (Iso.hom (α_ x x_1 x_2)))))
      Algebra.TensorProduct.includeRight)
    (AlgHom.comp
      (AlgHom.restrictScalars R
        (AlgCat.Hom.hom
          ((Iso.symm
                  (Iso.refl
                    (tensorObj (𝒞 := AlgCat.instCategory R)
                      (Functor.obj
                        (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                          carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                          (CD := AlgCat.carrier))
                        (tensorObj (𝒞 := category) x x_1))
                      (Functor.obj
                        (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                          carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                          (CD := AlgCat.carrier))
                        x_2))) ≪≫
                (Iso.symm
                    (Iso.refl
                      (tensorObj (𝒞 := AlgCat.instCategory R)
                        (Functor.obj
                          (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                            carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                            (CD := AlgCat.carrier))
                          x)
                        (Functor.obj
                          (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                            carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                            (CD := AlgCat.carrier))
                          x_1))) ⊗ᵢ
                  Iso.refl
                    (Functor.obj
                      (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                        carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD :=
                        AlgCat.carrier))
                      x_2))) ≪≫
              α_
                  (Functor.obj
                    (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                      carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD :=
                      AlgCat.carrier))
                    x)
                  (Functor.obj
                    (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                      carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD :=
                      AlgCat.carrier))
                    x_1)
                  (Functor.obj
                    (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                      carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD :=
                      AlgCat.carrier))
                    x_2) ≪≫
                (Iso.refl
                      (Functor.obj
                        (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                          carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                          (CD := AlgCat.carrier))
                        x) ⊗ᵢ
                    Iso.refl
                      (tensorObj (𝒞 := AlgCat.instCategory R)
                        (Functor.obj
                          (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                            carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                            (CD := AlgCat.carrier))
                          x_1)
                        (Functor.obj
                          (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                            carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                            (CD := AlgCat.carrier))
                          x_2))) ≪≫
                  Iso.refl
                    (tensorObj (𝒞 := AlgCat.instCategory R)
                      (Functor.obj
                        (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                          carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                          (CD := AlgCat.carrier))
                        x)
                      (Functor.obj
                        (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                          carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                          (CD := AlgCat.carrier))
                        (tensorObj (𝒞 := category) x_1 x_2)))).hom))
      Algebra.TensorProduct.includeRight)
have H : ∀ (x_3 : x_2.carrier),
  Eq (α :=
    (↑(Functor.obj
          (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC := carrier)
            (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
          (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))) :
      Type u))
    ((AlgHom.comp
            (AlgHom.restrictScalars R (S := R)
              (AlgCat.Hom.hom
                (CategoryTheory.Functor.map
                  (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC := carrier)
                    (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD :=
                    AlgCat.carrier))
                  (Iso.hom (α_ x x_1 x_2)))))
            Algebra.TensorProduct.includeRight :
          (a : x_2.carrier) → (↑((forget₂ (BialgCat R) (AlgCat R)).obj (x ⊗ x_1 ⊗ x_2)) : Type u))
        x_3 :
      (↑(Functor.obj
            (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC := carrier)
              (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
            (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))) :
        Type u))
    (((AlgHom.comp
              (AlgHom.restrictScalars R
                (AlgCat.Hom.hom
                  ((Iso.symm
                          (Iso.refl
                            (tensorObj (𝒞 := AlgCat.instCategory R)
                              (Functor.obj
                                (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                  (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                  (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                                (tensorObj (𝒞 := category) x x_1))
                              (Functor.obj
                                (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                  (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                  (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                                x_2))) ≪≫
                        (Iso.symm
                            (Iso.refl
                              (tensorObj (𝒞 := AlgCat.instCategory R)
                                (Functor.obj
                                  (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                    (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                    (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                                  x)
                                (Functor.obj
                                  (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                    (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                    (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                                  x_1))) ⊗ᵢ
                          Iso.refl
                            (Functor.obj
                              (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                              x_2))) ≪≫
                      α_
                          (Functor.obj
                            (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                              carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                              (CD := AlgCat.carrier))
                            x)
                          (Functor.obj
                            (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                              carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                              (CD := AlgCat.carrier))
                            x_1)
                          (Functor.obj
                            (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                              carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u))
                              (CD := AlgCat.carrier))
                            x_2) ≪≫
                        (Iso.refl
                              (Functor.obj
                                (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                  (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                  (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                                x) ⊗ᵢ
                            Iso.refl
                              (tensorObj (𝒞 := AlgCat.instCategory R)
                                (Functor.obj
                                  (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                    (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                    (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                                  x_1)
                                (Functor.obj
                                  (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                    (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                    (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                                  x_2))) ≪≫
                          Iso.refl
                            (tensorObj (𝒞 := AlgCat.instCategory R)
                              (Functor.obj
                                (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                  (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                  (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                                x)
                              (Functor.obj
                                (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier)
                                  (CC := carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦
                                  (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
                                (tensorObj (𝒞 := category) x_1 x_2)))).hom))
              Algebra.TensorProduct.includeRight :
            x_2.carrier →ₐ[R]
              (↑(Functor.obj
                    (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC :=
                      carrier) (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD :=
                      AlgCat.carrier))
                    (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))) :
                Type u)) :
          (a : x_2.carrier) → (↑((forget₂ (BialgCat R) (AlgCat R)).obj (x ⊗ x_1 ⊗ x_2)) : Type u))
        x_3 :
      (↑(Functor.obj
            (forget₂ (BialgCat R) (FC := fun (x1 x2 : BialgCat R) ↦ x1.carrier →ₐc[R] x2.carrier) (CC := carrier)
              (AlgCat R) (FD := fun (x1 x2 : AlgCat R) ↦ (↑x1 : Type u) →ₐ[R] (↑x2 : Type u)) (CD := AlgCat.carrier))
            (tensorObj (𝒞 := category) x (tensorObj (𝒞 := category) x_1 x_2))) :
        Type u))
Tactic: ext