-- 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) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w))
      (β := (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : 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) (tensorObj.{v, u_1} (𝒞 := inst) X X) 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) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))))) :
            (a : (X ⊗ X) ⊗ X ⟶ X) → ToType (F.obj (op ((X ⊗ X) ⊗ X))))
          (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) X 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 (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                      (tensorObj.{v, u_1} (𝒞 := inst) X X ⟶ X)) :
                  (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
                (HMul.hMul (α :=
                  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 X))))
                  (β :=
                  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 X))))
                  (((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 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 X))))) :
                        (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                      (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 X))))
                  (((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 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 X))))) :
                        (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                      (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 X))))) ▷
              X)
            (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) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))))
      (((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) (tensorObj.{v, u_1} (𝒞 := inst) X X) 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) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))))) :
            (a : (X ⊗ X) ⊗ X ⟶ X) → ToType (F.obj (op ((X ⊗ X) ⊗ X))))
          (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) X 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 (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                      (tensorObj.{v, u_1} (𝒞 := inst) X X ⟶ X)) :
                  (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
                (HMul.hMul (α :=
                  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 X))))
                  (β :=
                  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 X))))
                  (((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 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 X))))) :
                        (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                      (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 X))))
                  (((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 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 X))))) :
                        (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                      (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 X))))) ▷
              X)
            (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) (tensorObj.{v, u_1} (𝒞 := inst) X X) X)))) =
    HMul.hMul (α := (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w))
      (β := (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : 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) (tensorObj.{v, u_1} (𝒞 := inst) X X) 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) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))))) :
            (a : (X ⊗ X) ⊗ X ⟶ X) → ToType (F.obj (op ((X ⊗ X) ⊗ X))))
          (Iso.hom (α_ X X X) ≫
            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 (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                        (tensorObj.{v, u_1} (𝒞 := inst) X X ⟶ X)) :
                    (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
                  (HMul.hMul (α :=
                    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 X))))
                    (β :=
                    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 X))))
                    (((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 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 X))))) :
                          (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                        (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 X))))
                    (((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 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 X))))) :
                          (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                        (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 X))))) ≫
              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) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))))
      (((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) (tensorObj.{v, u_1} (𝒞 := inst) X X) 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) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))))) :
            (a : (X ⊗ X) ⊗ X ⟶ X) → ToType (F.obj (op ((X ⊗ X) ⊗ X))))
          (Iso.hom (α_ X X X) ≫
            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 (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                        (tensorObj.{v, u_1} (𝒞 := inst) X X ⟶ X)) :
                    (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
                  (HMul.hMul (α :=
                    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 X))))
                    (β :=
                    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 X))))
                    (((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 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 X))))) :
                          (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                        (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 X))))
                    (((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 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 X))))) :
                          (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                        (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 X))))) ≫
              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) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))))
⊢ HMul.hMul (α := (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w))
      (β := (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
              (X := F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X X))) (Y :=
              F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X)))
              (F.map
                (Quiver.Hom.op
                  (((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 (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                            (tensorObj.{v, u_1} (𝒞 := inst) X X ⟶ X)) :
                        (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
                      (HMul.hMul (α :=
                        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 X))))
                        (β :=
                        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 X))))
                        (((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 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 X))))) :
                              (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                            (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 X))))
                        (((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 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 X))))) :
                              (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                            (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 X))))) ▷
                    X))) :
            (a : (↑(F.obj (op (X ⊗ X))) : Type w)) → (↑(F.obj (op ((X ⊗ X) ⊗ X))) : 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 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 X))))) :
                (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
              (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 X)))) :
        (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
              (X := F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X X))) (Y :=
              F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X)))
              (F.map
                (Quiver.Hom.op
                  (((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 (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                            (tensorObj.{v, u_1} (𝒞 := inst) X X ⟶ X)) :
                        (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
                      (HMul.hMul (α :=
                        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 X))))
                        (β :=
                        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 X))))
                        (((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 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 X))))) :
                              (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                            (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 X))))
                        (((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 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 X))))) :
                              (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                            (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 X))))) ▷
                    X))) :
            (a : (↑(F.obj (op (X ⊗ X))) : Type w)) → (↑(F.obj (op ((X ⊗ X) ⊗ X))) : 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 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 X))))) :
                (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
              (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 X)))) :
        (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w)) =
    HMul.hMul (α := (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w))
      (β := (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
              (X := F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorObj.{v, u_1} (𝒞 := inst) X X))))
              (F.map (Quiver.Hom.op (Iso.hom (α_ X X X)))) :
            (a : (↑(F.obj (op (X ⊗ X ⊗ X))) : Type w)) → (↑(F.obj (op ((X ⊗ X) ⊗ X))) : Type w))
          ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC :=
                  MonCat.carrier) (X := F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X X))) (Y :=
                  F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                  (F.map
                    (Quiver.Hom.op
                      (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 (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                                (tensorObj.{v, u_1} (𝒞 := inst) X X ⟶ X)) :
                            (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
                          (HMul.hMul (α :=
                            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 X))))
                            (β :=
                            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 X))))
                            (((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 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 X))))) :
                                  (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                                (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 X))))
                            (((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 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 X))))) :
                                  (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                                (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 X)))))))) :
                (a : (↑(F.obj (op (X ⊗ X))) : Type w)) → (↑(F.obj (op (X ⊗ X ⊗ X))) : 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 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 X))))) :
                    (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                  (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 X)))) :
            (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorObj.{v, u_1} (𝒞 := inst) X X)))) : Type w)) :
        (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC := MonCat.carrier)
              (X := F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorObj.{v, u_1} (𝒞 := inst) X X))))
              (F.map (Quiver.Hom.op (Iso.hom (α_ X X X)))) :
            (a : (↑(F.obj (op (X ⊗ X ⊗ X))) : Type w)) → (↑(F.obj (op ((X ⊗ X) ⊗ X))) : Type w))
          ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{w}) ↦ (↑x1 : Type w) →* (↑x2 : Type w)) (CC :=
                  MonCat.carrier) (X := F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X X))) (Y :=
                  F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                  (F.map
                    (Quiver.Hom.op
                      (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 (tensorObj.{v, u_1} (𝒞 := inst) X X))))
                                (tensorObj.{v, u_1} (𝒞 := inst) X X ⟶ X)) :
                            (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
                          (HMul.hMul (α :=
                            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 X))))
                            (β :=
                            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 X))))
                            (((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 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 X))))) :
                                  (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                                (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 X))))
                            (((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 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 X))))) :
                                  (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                                (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 X)))))))) :
                (a : (↑(F.obj (op (X ⊗ X))) : Type w)) → (↑(F.obj (op (X ⊗ X ⊗ X))) : 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 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 X))))) :
                    (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
                  (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 X)))) :
            (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) X (tensorObj.{v, u_1} (𝒞 := inst) X X)))) : Type w)) :
        (↑(F.obj (op (tensorObj.{v, u_1} (𝒞 := inst) (tensorObj.{v, u_1} (𝒞 := inst) X X) X))) : Type w))

Tactic:
simp only [← α.homEquiv'_comp]

Premises used: CategoryTheory.Functor.RepresentableBy.homEquiv'_comp