-- Module: Mathlib.CategoryTheory.Monoidal.Cartesian.FunctorCategory
Goal:
J : Type u_1
C : Type u_2
D : Type u_3
E : Type u_4
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : Category.{v_3, u_3} D
inst_3 : Category.{v_4, u_4} E
inst_4 : CartesianMonoidalCategory.{v_2, u_2} C
inst_5 : CartesianMonoidalCategory.{v_4, u_4} E
F G : Functor.{v_2, v, u_2, v + 1} Cᵒᵖ (Type v)
X Y : C
h₁ : F.RepresentableBy X
h₂ : G.RepresentableBy Y
I W : C
f : I ⟶ W
g : W ⟶ tensorObj.{v_2, u_2} (𝒞 := inst_1) X Y
refine_1 :
  Eq.{v + 1} (α := F.obj (Opposite.op I))
    (((Equiv.trans homEquivToProd (Equiv.prodCongr h₁.homEquiv h₂.homEquiv) :
            Equiv.{v_2 + 1, v + 1} (I ⟶ tensorObj.{v_2, u_2} (𝒞 := inst_1) X Y)
              ((tensorObj (𝒞 := category) F G).obj (Opposite.op I))) :
          (a : I ⟶ X ⊗ Y) → (F ⊗ G).obj (Opposite.op I))
        (f ≫ g)).1
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((tensorObj (𝒞 := category) F G).map f.op) :
          (a : (fun (X : Type v) ↦ X) ((F ⊗ G).obj (Opposite.op W))) →
            (fun (X : Type v) ↦ X) ((F ⊗ G).obj (Opposite.op I)))
        (((Equiv.trans homEquivToProd (Equiv.prodCongr h₁.homEquiv h₂.homEquiv) :
                Equiv.{v_2 + 1, v + 1} (W ⟶ tensorObj.{v_2, u_2} (𝒞 := inst_1) X Y)
                  ((tensorObj (𝒞 := category) F G).obj (Opposite.op W))) :
              (a : W ⟶ X ⊗ Y) → (F ⊗ G).obj (Opposite.op W))
            g :
          (tensorObj (𝒞 := category) F G).obj (Opposite.op W))).1
refine_2 :
  Eq.{v + 1} (α := G.obj (Opposite.op I))
    (((Equiv.trans homEquivToProd (Equiv.prodCongr h₁.homEquiv h₂.homEquiv) :
            Equiv.{v_2 + 1, v + 1} (I ⟶ tensorObj.{v_2, u_2} (𝒞 := inst_1) X Y)
              ((tensorObj (𝒞 := category) F G).obj (Opposite.op I))) :
          (a : I ⟶ X ⊗ Y) → (F ⊗ G).obj (Opposite.op I))
        (f ≫ g)).2
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((tensorObj (𝒞 := category) F G).map f.op) :
          (a : (fun (X : Type v) ↦ X) ((F ⊗ G).obj (Opposite.op W))) →
            (fun (X : Type v) ↦ X) ((F ⊗ G).obj (Opposite.op I)))
        (((Equiv.trans homEquivToProd (Equiv.prodCongr h₁.homEquiv h₂.homEquiv) :
                Equiv.{v_2 + 1, v + 1} (W ⟶ tensorObj.{v_2, u_2} (𝒞 := inst_1) X Y)
                  ((tensorObj (𝒞 := category) F G).obj (Opposite.op W))) :
              (a : W ⟶ X ⊗ Y) → (F ⊗ G).obj (Opposite.op W))
            g :
          (tensorObj (𝒞 := category) F G).obj (Opposite.op W))).2
⊢ Eq.{v + 1} (α := (tensorObj (𝒞 := category) F G).obj (Opposite.op I))
    (((Equiv.trans homEquivToProd (Equiv.prodCongr h₁.homEquiv h₂.homEquiv) :
            Equiv.{v_2 + 1, v + 1} (I ⟶ tensorObj.{v_2, u_2} (𝒞 := inst_1) X Y)
              ((tensorObj (𝒞 := category) F G).obj (Opposite.op I))) :
          (a : I ⟶ X ⊗ Y) → (F ⊗ G).obj (Opposite.op I))
        (f ≫ g) :
      (tensorObj (𝒞 := category) F G).obj (Opposite.op I))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((tensorObj (𝒞 := category) F G).map f.op) :
          (a : (fun (X : Type v) ↦ X) ((F ⊗ G).obj (Opposite.op W))) →
            (fun (X : Type v) ↦ X) ((F ⊗ G).obj (Opposite.op I)))
        (((Equiv.trans homEquivToProd (Equiv.prodCongr h₁.homEquiv h₂.homEquiv) :
                Equiv.{v_2 + 1, v + 1} (W ⟶ tensorObj.{v_2, u_2} (𝒞 := inst_1) X Y)
                  ((tensorObj (𝒞 := category) F G).obj (Opposite.op W))) :
              (a : W ⟶ X ⊗ Y) → (F ⊗ G).obj (Opposite.op W))
            g :
          (tensorObj (𝒞 := category) F G).obj (Opposite.op W)) :
      (fun (X : Type v) ↦ X) ((tensorObj (𝒞 := category) F G).obj (Opposite.op I)))

Tactic:
refine Prod.ext ?_ ?_

Premises used: Prod.ext