-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F F' F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G' : Subfunctor F
f : G.toFunctor ⟶ F'
h : Presieve.IsSheaf J F'
U V : Cᵒᵖ
i : U ⟶ V
s : (sheafify J G).toFunctor.obj U
W : C
j : W ⟶ unop V
hj :
  Sieve.arrows
    (G.sieveOfSection
      (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((sheafify J G).toFunctor.map i) :
              (a : (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj U)) →
                (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj V))
            s) :
        F.obj V))
    j
⊢ Eq (α := (fun (X : Type w) ↦ X) (F'.obj (op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.map j.op) :
          (a : (fun (X : Type w) ↦ X) (F'.obj (op (unop V)))) → (fun (X : Type w) ↦ X) (F'.obj (op W)))
        (TypeCat.Fun.toFun
          (ConcreteCategory.hom (FC := TypeCat.Fun)
            ((sheafify J G).toFunctor.map i ≫
              ↾fun (s : (sheafify J G).toFunctor.obj V) ↦
                ⋯.amalgamate (R := (G.sieveOfSection (↑s : F.obj V)).arrows)
                  ((G.familyOfElementsOfSection (↑s : F.obj V)).map (R := (G.sieveOfSection (↑s : F.obj V)).arrows) f)
                  ⋯))
          s) :
      (fun (X : Type w) ↦ X) (F'.obj (op W)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F'.map j.op) :
          (a : (fun (X : Type w) ↦ X) (F'.obj (op (unop V)))) → (fun (X : Type w) ↦ X) (F'.obj (op W)))
        (TypeCat.Fun.toFun
          (ConcreteCategory.hom (FC := TypeCat.Fun)
            ((↾fun (s : (sheafify J G).toFunctor.obj U) ↦
                ⋯.amalgamate (R := (G.sieveOfSection (↑s : F.obj U)).arrows)
                  ((G.familyOfElementsOfSection (↑s : F.obj U)).map (R := (G.sieveOfSection (↑s : F.obj U)).arrows) f)
                  ⋯) ≫
              F'.map i))
          s) :
      (fun (X : Type w) ↦ X) (F'.obj (op W)))
have h₁ : Presieve.FamilyOfElements.map (R :=
    Sieve.arrows
      (G.sieveOfSection
        (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((sheafify J G).toFunctor.map i) :
                (a : (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj U)) →
                  (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj V))
              s) :
          F.obj V)))
    (G.familyOfElementsOfSection
      (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((sheafify J G).toFunctor.map i) :
              (a : (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj U)) →
                (fun (X : Type w) ↦ X) ((sheafify J G).toFunctor.obj V))
            s) :
        F.obj V))
    f j hj =
  (ConcreteCategory.hom (FC := TypeCat.Fun) (F'.map j.op) :
      (a : (fun (X : Type w) ↦ X) (F'.obj (op (unop V)))) → (fun (X : Type w) ↦ X) (F'.obj (op W)))
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((↾fun (s : (sheafify J G).toFunctor.obj U) ↦
            sorry.amalgamate (R := (G.sieveOfSection (↑s : F.obj U)).arrows)
              ((G.familyOfElementsOfSection (↑s : F.obj U)).map (R := (G.sieveOfSection (↑s : F.obj U)).arrows) f) sorry) ≫
          F'.map i))
      s)
Tactic: refine (Presieve.IsSheafFor.valid_glue (h _ ((G.sheafify J).toFunctor.map i s).2)
      ((G.family_of_elements_compatible _).map _) _ hj).trans ?_