-- Module: Mathlib.CategoryTheory.Topos.Sheaf
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F G : Cᵒᵖ ⥤ Type (max u v)
m : F ⟶ G
inst_1 : Mono m
hF : Presieve.IsSheaf J F
hG : Presieve.IsSeparated J G
X : Cᵒᵖ
x : G.obj X
Y : C
f : Y ⟶ Opposite.unop X
hf :
  J.Covers
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (↾fun (x : G.obj X) ↦
            {
              arrows := fun (Y : C) (f : Y ⟶ Opposite.unop X) ↦
                ∃ (a : F.obj (Opposite.op Y)),
                  Eq (α := G.obj (Opposite.op Y))
                    ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f.op) : (a : G.obj X) → G.obj (Opposite.op Y)) x :
                      G.obj (Opposite.op Y))
                    ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app (Opposite.op Y)) :
                          (a : F.obj (Opposite.op Y)) → G.obj (Opposite.op Y))
                        a :
                      G.obj (Opposite.op Y)),
              downward_closed := ⋯ : Sieve.{v, u} (Opposite.unop X) }) :
        (a : G.obj X) → (Functor.sieves C).obj X)
      x)
    f
a :
  (Z : C) →
    (g : Z ⟶ Y) →
      (hg :
          (Sieve.pullback f
                ((ConcreteCategory.hom (FC := TypeCat.Fun) ((χ m).app X) :
                    (a : (fun (X : Type (max u v)) ↦ X) (G.obj X)) →
                      (fun (X : Type (max u v)) ↦ X) ((Functor.sieves C).obj X))
                  x)).arrows
            g) →
        (fun (X : Type (max u v)) ↦ X) (F.obj (Opposite.op Z))
ha :
  ∀ (Z : C) (g : Z ⟶ Y)
    (hg :
      (Sieve.pullback f
            ((ConcreteCategory.hom (FC := TypeCat.Fun) ((χ m).app X) :
                (a : (fun (X : Type (max u v)) ↦ X) (G.obj X)) →
                  (fun (X : Type (max u v)) ↦ X) ((Functor.sieves C).obj X))
              x)).arrows
        g),
    Eq (α := (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map (g ≫ f).op) :
            (a : (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op (Opposite.unop X)))) →
              (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
          x :
        (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app (Opposite.op Z)) :
            (a : (fun (X : Type (max u v)) ↦ X) (F.obj (Opposite.op Z))) →
              (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
          (a Z g hg) :
        (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
refine_2 :
  ∀ (Z : C) (f' : Z ⟶ Y)
    (hf' :
      (Sieve.pullback f
            ((ConcreteCategory.hom (FC := TypeCat.Fun)
                  (↾fun (x : G.obj X) ↦
                    {
                      arrows := fun (Y : C) (f : Y ⟶ Opposite.unop X) ↦
                        ∃ (a : F.obj (Opposite.op Y)),
                          Eq (α := G.obj (Opposite.op Y))
                            ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f.op) :
                                  (a : G.obj X) → G.obj (Opposite.op Y))
                                x :
                              G.obj (Opposite.op Y))
                            ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app (Opposite.op Y)) :
                                  (a : F.obj (Opposite.op Y)) → G.obj (Opposite.op Y))
                                a :
                              G.obj (Opposite.op Y)),
                      downward_closed := ⋯ : Sieve.{v, u} (Opposite.unop X) }) :
                (a : G.obj X) → (Functor.sieves C).obj X)
              x)).arrows
        f'),
    Eq (α := (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f'.op) :
            (a : (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Y))) →
              (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f.op) : (a : G.obj X) → G.obj (Opposite.op Y)) x) :
        (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f'.op) :
            (a : (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Y))) →
              (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app (Opposite.op Y)) :
              (a : F.obj (Opposite.op Y)) → G.obj (Opposite.op Y))
            (⋯.amalgamate (R :=
              (Sieve.pullback f
                  ((ConcreteCategory.hom (FC := TypeCat.Fun)
                        (↾fun (x : G.obj X) ↦
                          {
                            arrows := fun (Y : C) (f : Y ⟶ Opposite.unop X) ↦
                              ∃ (a : F.obj (Opposite.op Y)),
                                Eq (α := G.obj (Opposite.op Y))
                                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f.op) :
                                        (a : G.obj X) → G.obj (Opposite.op Y))
                                      x :
                                    G.obj (Opposite.op Y))
                                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app (Opposite.op Y)) :
                                        (a : F.obj (Opposite.op Y)) → G.obj (Opposite.op Y))
                                      a :
                                    G.obj (Opposite.op Y)),
                            downward_closed := ⋯ : Sieve.{v, u} (Opposite.unop X) }) :
                      (a : G.obj X) → (Functor.sieves C).obj X)
                    x)).arrows)
              a ⋯)) :
        (fun (X : Type (max u v)) ↦ X) (G.obj (Opposite.op Z)))
⊢ Eq (α := G.obj (Opposite.op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f.op) : (a : G.obj X) → G.obj (Opposite.op Y)) x :
      G.obj (Opposite.op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app (Opposite.op Y)) :
          (a : F.obj (Opposite.op Y)) → G.obj (Opposite.op Y))
        (⋯.amalgamate (R :=
          (Sieve.pullback f
              ((ConcreteCategory.hom (FC := TypeCat.Fun)
                    (↾fun (x : G.obj X) ↦
                      {
                        arrows := fun (Y : C) (f : Y ⟶ Opposite.unop X) ↦
                          ∃ (a : F.obj (Opposite.op Y)),
                            Eq (α := G.obj (Opposite.op Y))
                              ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f.op) :
                                    (a : G.obj X) → G.obj (Opposite.op Y))
                                  x :
                                G.obj (Opposite.op Y))
                              ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app (Opposite.op Y)) :
                                    (a : F.obj (Opposite.op Y)) → G.obj (Opposite.op Y))
                                  a :
                                G.obj (Opposite.op Y)),
                        downward_closed := ⋯ : Sieve.{v, u} (Opposite.unop X) }) :
                  (a : G.obj X) → (Functor.sieves C).obj X)
                x)).arrows)
          a ⋯) :
      G.obj (Opposite.op Y))

Tactic:
refine (hG _ hf).ext fun Z f' hf' => ?_

Premises used: CategoryTheory.Presieve.IsSeparatedFor.ext