-- Module: Mathlib.CategoryTheory.Sites.IsSheafFor
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) }
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)
                          ⋯ :
                        { 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)) :
          FamilyOfElements F S.arrows)
        f hf =
      (↑x : FamilyOfElements F S.arrows) f hf
⊢ (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

Tactic:
ext