-- Module: Mathlib.CategoryTheory.Sites.Subsheaf
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F_1 F'_1 F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G'_1 : Subfunctor F_1
F F' : Sheaf J (Type w)
f : F ⟶ F'
G' : Sheaf J (Type w)
g₁ g₂ : Sheaf.image f ⟶ G'
e : Sheaf.toImage f ≫ g₁ = Sheaf.toImage f ≫ g₂
U : Cᵒᵖ
s : F'.obj (P := Presheaf.IsSheaf J).obj U
hx : s ∈ (Subfunctor.sheafify J (Subfunctor.range f.hom)).obj U
V : C
i : V ⟶ unop U
y : F.obj (P := Presheaf.IsSheaf J).obj (op V)
e' :
  Eq (α := F'.obj (P := Presheaf.IsSheaf J).obj (op V))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.hom.app (op V)) :
          (a : (fun (X : Type w) ↦ X) (F.obj.obj (op V))) → (fun (X : Type w) ↦ X) (F'.obj.obj (op V)))
        y :
      (fun (X : Type w) ↦ X) (F'.obj (P := Presheaf.IsSheaf J).obj (op V)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.obj (P := Presheaf.IsSheaf J).map i.op) :
          (a : (fun (X : Type w) ↦ X) (F'.obj.obj (op (unop U)))) → (fun (X : Type w) ↦ X) (F'.obj.obj (op V)))
        s :
      (fun (X : Type w) ↦ X) (F'.obj (P := Presheaf.IsSheaf J).obj (op V)))
this :
  Eq (α := G'.obj (P := Presheaf.IsSheaf J).obj (op V))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (g₁.hom.app (op V)) :
          (a : (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w)) → G'.obj.obj (op V))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Subfunctor.toRangeSheafify J f.hom).app (op V)) :
              (a : F.obj.obj (op V)) → (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w))
            y :
          (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w)) :
      G'.obj (P := Presheaf.IsSheaf J).obj (op V))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (g₂.hom.app (op V)) :
          (a : (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w)) → G'.obj.obj (op V))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((Subfunctor.toRangeSheafify J f.hom).app (op V)) :
              (a : F.obj.obj (op V)) → (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w))
            y :
          (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w)) :
      G'.obj (P := Presheaf.IsSheaf J).obj (op V))
E :
  Eq (α := F'.obj (P := Presheaf.IsSheaf J).obj (op V))
    ((↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((Subfunctor.toRangeSheafify J f.hom).app (op V)) :
              (a : F.obj.obj (op V)) → (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w))
            y) :
        F'.obj (P := Presheaf.IsSheaf J).obj (op V)) :
      F'.obj (P := Presheaf.IsSheaf J).obj (op V))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.obj (P := Presheaf.IsSheaf J).map i.op) :
          (a : F'.obj.obj U) → F'.obj.obj (op V))
        s :
      F'.obj (P := Presheaf.IsSheaf J).obj (op V))
h h₁ :
  Eq (α := F'.obj (P := Presheaf.IsSheaf J).obj (op V))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.obj (P := Presheaf.IsSheaf J).map i.op) :
          (a : F'.obj.obj U) → F'.obj.obj (op V))
        s :
      F'.obj (P := Presheaf.IsSheaf J).obj (op V))
    ((↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((Subfunctor.toRangeSheafify J f.hom).app (op V)) :
              (a : F.obj.obj (op V)) → (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w))
            y) :
        F'.obj (P := Presheaf.IsSheaf J).obj (op V)) :
      F'.obj (P := Presheaf.IsSheaf J).obj (op V))
⊢ Eq (α := G'.obj (P := Presheaf.IsSheaf J).obj (op V))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (g₁.hom.app (op V)) :
          (a : (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w)) → G'.obj.obj (op V))
        (Subtype.mk (p := fun (x : F'.obj (P := Presheaf.IsSheaf J).obj (op V)) ↦
            Membership.mem (γ := Set (F'.obj (P := Presheaf.IsSheaf J).obj (op V)))
              ((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) x)
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.obj (P := Presheaf.IsSheaf J).map i.op) :
                (a : F'.obj.obj U) → F'.obj.obj (op V))
              s)
            ⋯ :
          { x : F'.obj (P := Presheaf.IsSheaf J).obj (op V) //
            Membership.mem (γ := Set (F'.obj (P := Presheaf.IsSheaf J).obj (op V)))
              ((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) x }) :
      G'.obj (P := Presheaf.IsSheaf J).obj (op V))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (g₂.hom.app (op V)) :
          (a : (↑((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) : Type w)) → G'.obj.obj (op V))
        (Subtype.mk (p := fun (x : F'.obj (P := Presheaf.IsSheaf J).obj (op V)) ↦
            Membership.mem (γ := Set (F'.obj (P := Presheaf.IsSheaf J).obj (op V)))
              ((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) x)
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.obj (P := Presheaf.IsSheaf J).map i.op) :
                (a : F'.obj.obj U) → F'.obj.obj (op V))
              s)
            ⋯ :
          { x : F'.obj (P := Presheaf.IsSheaf J).obj (op V) //
            Membership.mem (γ := Set (F'.obj (P := Presheaf.IsSheaf J).obj (op V)))
              ((Subfunctor.sheafify J (Subfunctor.range f.hom)).obj (op V)) x }) :
      G'.obj (P := Presheaf.IsSheaf J).obj (op V))

Tactic:
convert! this