-- Module: Mathlib.Topology.Category.TopCat.GrothendieckTopology
Goal:
X Y : TopCat
𝒰 : precoverage.ZeroHypercover Y
heq :
  ∀ (i : 𝒰.I₀) (y : (↑(𝒰.X i) : Type u_1)),
    Eq (α := (↑Y : Type u_1))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (𝒰.f i) :
            (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1))
          y :
        (↑Y : Type u_1))
      ((↑(Subtype.mk (p := fun (x : (↑Y : Type u_1)) ↦
              Membership.mem (γ := Set (↑Y : Type u_1))
                (Set.range (ι := (↑(𝒰.X i) : Type u_1))
                  (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                        carrier) (𝒰.f i)) :
                    (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1)))
                x)
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                    carrier) (𝒰.f i) :
                  (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1))
                y)
              ⋯) :
          (↑Y : Type u_1)) :
        (↑Y : Type u_1))
refine_1 :
  (x : (i : 𝒰.I₀) → Functor.obj (Functor.obj yoneda X) (Opposite.op (𝒰.X i))) →
    (hx : Presieve.Arrows.Compatible (Functor.obj yoneda X) 𝒰.f x) →
      (fun (X : Type u_1) ↦ X) (Functor.obj (Functor.obj yoneda X) (Opposite.op Y))
refine_2 :
  ∀ (x : (i : 𝒰.I₀) → Functor.obj (Functor.obj yoneda X) (Opposite.op (𝒰.X i)))
    (hx : Presieve.Arrows.Compatible (Functor.obj yoneda X) 𝒰.f x) (i : 𝒰.I₀),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (CategoryTheory.Functor.map (Functor.obj yoneda X) (𝒰.f i).op) :
          (a : (fun (X : Type u_1) ↦ X) ((yoneda.obj X).obj (Opposite.op Y))) →
            (fun (X : Type u_1) ↦ X) ((yoneda.obj X).obj (Opposite.op (𝒰.X i))))
        (ofHom
          (ContinuousMap.liftCover
            (fun (i : 𝒰.I₀) ↦
              Set.range (ι := (↑(𝒰.X i) : Type u_1))
                (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                      carrier) (𝒰.f i)) :
                  (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1)))
            (fun (i : 𝒰.I₀) ↦
              {
                toFun :=
                  Function.comp (β := (↑(Opposite.unop (Opposite.op (𝒰.X i))) : Type u_1))
                    (⇑(Hom.hom (x i)) : (a : (↑(Opposite.unop (Opposite.op (𝒰.X i))) : Type u_1)) → (↑X : Type u_1))
                    (⇑⋯.toHomeomorph.symm :
                      (a :
                          (↑(Set.range
                                (⇑(ConcreteCategory.hom (𝒰.f i)) : (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1))) :
                            Type u_1)) →
                        (↑(𝒰.X i) : Type u_1)),
                continuous_toFun := ⋯ })
            ⋯ ⋯)) =
      x i
refine_3 :
  ∀ (x : (i : 𝒰.I₀) → Functor.obj (Functor.obj yoneda X) (Opposite.op (𝒰.X i)))
    (hx : Presieve.Arrows.Compatible (Functor.obj yoneda X) 𝒰.f x)
    (f : (fun (X : Type u_1) ↦ X) (Functor.obj (Functor.obj yoneda X) (Opposite.op Y)))
    (hf :
      (fun (t : (fun (X : Type u_1) ↦ X) (Functor.obj (Functor.obj yoneda X) (Opposite.op Y))) ↦
          ∀ (i : 𝒰.I₀),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (CategoryTheory.Functor.map (Functor.obj yoneda X) (𝒰.f i).op) :
                  (a : (fun (X : Type u_1) ↦ X) ((yoneda.obj X).obj (Opposite.op Y))) →
                    (fun (X : Type u_1) ↦ X) ((yoneda.obj X).obj (Opposite.op (𝒰.X i))))
                t =
              x i)
        f),
    f =
      ofHom
        (ContinuousMap.liftCover
          (fun (i : 𝒰.I₀) ↦
            Set.range (ι := (↑(𝒰.X i) : Type u_1))
              (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                    carrier) (𝒰.f i)) :
                (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1)))
          (fun (i : 𝒰.I₀) ↦
            {
              toFun :=
                Function.comp (β := (↑(Opposite.unop (Opposite.op (𝒰.X i))) : Type u_1))
                  (⇑(Hom.hom (x i)) : (a : (↑(Opposite.unop (Opposite.op (𝒰.X i))) : Type u_1)) → (↑X : Type u_1))
                  (⇑⋯.toHomeomorph.symm :
                    (a :
                        (↑(Set.range
                              (⇑(ConcreteCategory.hom (𝒰.f i)) : (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1))) :
                          Type u_1)) →
                      (↑(𝒰.X i) : Type u_1)),
              continuous_toFun := ⋯ })
          ⋯ ⋯)
⊢ ∀ (x : (i : 𝒰.I₀) → Functor.obj (Functor.obj yoneda X) (Opposite.op (𝒰.X i))),
    Presieve.Arrows.Compatible (Functor.obj yoneda X) 𝒰.f x →
      ∃! t : (fun (X : Type u_1) ↦ X) (Functor.obj (Functor.obj yoneda X) (Opposite.op Y)),
        ∀ (i : 𝒰.I₀),
          (ConcreteCategory.hom (FC := TypeCat.Fun) (CategoryTheory.Functor.map (Functor.obj yoneda X) (𝒰.f i).op) :
                (a : (fun (X : Type u_1) ↦ X) ((yoneda.obj X).obj (Opposite.op Y))) →
                  (fun (X : Type u_1) ↦ X) ((yoneda.obj X).obj (Opposite.op (𝒰.X i))))
              t =
            x i

Tactic:
refine fun x hx ↦ ⟨?_, fun i ↦ ?_, fun f hf ↦ ?_⟩