-- Module: Mathlib.CategoryTheory.Sites.IsSheafFor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
S : Sieve.{v₁, u₁} X
inst_1 : LocallySmall.{w, v₁, u₁} C
F : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
f : Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S) ⟶ F
g : Functor.obj shrinkYoneda.{w, v₁, u₁} X ⟶ F
refine_1 :
  Subfunctor.ι (shrinkFunctor.{w, v₁, u₁} S) ≫ g = f →
    ∀ ⦃Y : C⦄ (f_1 : Y ⟶ X) (h : S.arrows f_1),
      Eq (α := F.obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f_1.op) : (a : F.obj (op X)) → F.obj (op Y))
            ((shrinkYonedaEquiv : (a : shrinkYoneda.{w, v₁, u₁}.obj X ⟶ F) → F.obj (op X)) g : F.obj (op X)) :
          F.obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op Y)) :
              (a :
                  { x : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y) //
                    S.arrows
                      ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) 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)) ↦
                S.arrows (Y := Y)
                  ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x))
                ((Equiv.symm shrinkYonedaObjObjEquiv : (a : Y ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) f_1)
                ⋯ :
              { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
                S.arrows (Y := Y)
                  ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x) }) :
          F.obj (op Y))
refine_2 :
  ∀
    (h :
      ∀ ⦃Y : C⦄ (f_1 : Y ⟶ X) (h : S.arrows f_1),
        Eq (α := F.obj (op Y))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f_1.op) : (a : F.obj (op X)) → F.obj (op Y))
              ((shrinkYonedaEquiv : (a : shrinkYoneda.{w, v₁, u₁}.obj X ⟶ F) → F.obj (op X)) g : F.obj (op X)) :
            F.obj (op Y))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op Y)) :
                (a :
                    { x : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y) //
                      S.arrows
                        ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) 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)) ↦
                  S.arrows (Y := Y)
                    ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x))
                  ((Equiv.symm shrinkYonedaObjObjEquiv : (a : Y ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) f_1)
                  ⋯ :
                { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
                  S.arrows (Y := Y)
                    ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x) }) :
            F.obj (op Y))),
    Subfunctor.ι (shrinkFunctor.{w, v₁, u₁} S) ≫ g = f
⊢ Subfunctor.ι (shrinkFunctor.{w, v₁, u₁} S) ≫ g = f ↔
    ∀ ⦃Y : C⦄ (f_1 : Y ⟶ X) (h : S.arrows f_1),
      Eq (α := F.obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f_1.op) : (a : F.obj (op X)) → F.obj (op Y))
            ((shrinkYonedaEquiv : (a : shrinkYoneda.{w, v₁, u₁}.obj X ⟶ F) → F.obj (op X)) g : F.obj (op X)) :
          F.obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op Y)) :
              (a :
                  { x : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y) //
                    S.arrows
                      ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) 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)) ↦
                S.arrows (Y := Y)
                  ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x))
                ((Equiv.symm shrinkYonedaObjObjEquiv : (a : Y ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) f_1)
                ⋯ :
              { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
                S.arrows (Y := Y)
                  ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x) }) :
          F.obj (op Y))

Tactic:
refine ⟨?_, fun h ↦ ?_⟩