-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P Q U : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X Y : C
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
inst_1 : LocallySmall.{w, v₁, u₁} C
F : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
x : { x : FamilyOfElements F S.arrows // x.Compatible (R := S.arrows) }
⊢ (fun (t : Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S) ⟶ F) ↦
        Subtype.mk (p := fun (x : FamilyOfElements F S.arrows) ↦ x.Compatible (R := S.arrows))
          (fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun) (t.app (op Y)) :
                (a : (fun (X : Type w) ↦ X) ((shrinkFunctor.{w, v₁, u₁} S).toFunctor.obj (op Y))) →
                  (fun (X : Type w) ↦ X) (F.obj (op Y)))
              (Subtype.mk (p :=
                  fun (x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)) ↦
                  Membership.mem (γ :=
                    Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)))
                    (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) (op Y)) x)
                  ((Equiv.symm shrinkYonedaObjObjEquiv :
                      (a : unop (op Y) ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y))
                    f)
                  ⋯ :
                { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
                  Membership.mem (γ :=
                    Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)))
                    (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) (op Y)) x }))
          ⋯)
      ((fun (t : { x : FamilyOfElements F S.arrows // x.Compatible (R := S.arrows) }) ↦
          ({
              app := fun (X_1 : Cᵒᵖ) ↦
                ↾fun
                    (f :
                      (Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X)
                            (shrinkFunctor.{w, v₁, u₁} S)).obj
                        X_1) ↦
                  (↑t : FamilyOfElements F S.arrows)
                    ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj X_1) → unop X_1 ⟶ X)
                      (↑f : (shrinkYoneda.{w, v₁, u₁}.obj X).obj X_1))
                    ⋯,
              naturality := ⋯ } :
            NatTrans (Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S))
              F))
        x) =
    x
have a : ∀ {Y : C} (f : Y ⟶ X) (hf : S.arrows f),
  (↑((fun (t : Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S) ⟶ F) ↦
              Subtype.mk (p := fun (x : FamilyOfElements F S.arrows) ↦ x.Compatible (R := S.arrows))
                (fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦
                  (ConcreteCategory.hom (FC := TypeCat.Fun) (t.app (op Y)) :
                      (a : (fun (X : Type w) ↦ X) ((shrinkFunctor.{w, v₁, u₁} S).toFunctor.obj (op Y))) →
                        (fun (X : Type w) ↦ X) (F.obj (op Y)))
                    (Subtype.mk (p :=
                        fun (x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)) ↦
                        Membership.mem (γ :=
                          Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)))
                          (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) (op Y)) x)
                        ((Equiv.symm shrinkYonedaObjObjEquiv :
                            (a : unop (op Y) ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y))
                          f)
                        sorry :
                      { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
                        Membership.mem (γ :=
                          Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)))
                          (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) (op Y)) x }))
                sorry)
            ((fun (t : { x : FamilyOfElements F S.arrows // x.Compatible (R := S.arrows) }) ↦
                ({
                    app := fun (X_1 : Cᵒᵖ) ↦
                      ↾fun
                          (f :
                            (Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X)
                                  (shrinkFunctor.{w, v₁, u₁} S)).obj
                              X_1) ↦
                        (↑t : FamilyOfElements F S.arrows)
                          ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj X_1) → unop X_1 ⟶ X)
                            (↑f : (shrinkYoneda.{w, v₁, u₁}.obj X).obj X_1))
                          sorry,
                    naturality := sorry } :
                  NatTrans
                    (Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S))
                    F))
              x)) :
        FamilyOfElements F S.arrows)
      f hf =
    (↑x : FamilyOfElements F S.arrows) f hf
Tactic: ext