-- Module: Mathlib.CategoryTheory.Sites.IsSheafFor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
S : Sieve.{v₁, u₁} X
P : Functor.{v₁, v₁, u₁, v₁ + 1} Cᵒᵖ (Type v₁)
a : S.functor ⟶ P
b : Functor.obj yoneda X ⟶ P
a_1 : Cᵒᵖ
a₁ :
  (∀
      (x :
        { x : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) a_1 //
          S.arrows (Y := unop a_1)
            ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj a_1) → unop a_1 ⟶ X) x) }),
      Eq (α := P.obj a_1)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (b.app a_1) : (a : unop a_1 ⟶ X) → P.obj a_1)
            ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj a_1) → unop a_1 ⟶ X)
                (↑x : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj a_1) :
              unop a_1 ⟶ X) :
          P.obj a_1)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (a.app a_1) : (a : { g : unop a_1 ⟶ X // S.arrows g }) → P.obj a_1)
            ((Equiv.subtypeEquiv (p :=
                    Membership.mem (γ := Set (Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) a_1))
                      {f : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) a_1 |
                        S.arrows (Y := unop a_1)
                          ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj a_1) → unop a_1 ⟶ X)
                            f)})
                    (q := S.arrows) shrinkYonedaObjObjEquiv ⋯ :
                  (a :
                      { x : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj a_1 //
                        S.arrows
                          ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj a_1) → unop a_1 ⟶ X)
                            x) }) →
                    { g : unop a_1 ⟶ X // S.arrows g })
                x :
              { g : unop a_1 ⟶ X // S.arrows g }) :
          P.obj a_1)) ↔
    ∀ (x : { g : unop a_1 ⟶ X // S.arrows g }),
      Eq (α := P.obj a_1)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (b.app a_1) : (a : unop a_1 ⟶ X) → P.obj a_1) (↑x : unop a_1 ⟶ X) :
          P.obj a_1)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (a.app a_1) : (a : { g : unop a_1 ⟶ X // S.arrows g }) → P.obj a_1)
            x :
          P.obj a_1)
⊢ (∀ (x : (Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) (shrinkFunctor.{v₁, v₁, u₁} S)).obj a_1),
      Eq (α := P.obj a_1)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app
                  (Subfunctor.ι (shrinkFunctor.{v₁, v₁, u₁} S) ≫
                    Iso.hom (NatIso.ofComponents (fun (Y : Cᵒᵖ) ↦ Equiv.toIso shrinkYonedaObjObjEquiv) ⋯) ≫ b)
                  a_1) :
              (a : (shrinkFunctor.{v₁, v₁, u₁} S).toFunctor.obj a_1) → P.obj a_1)
            x :
          P.obj a_1)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((S.shrinkFunctorIsoFunctor.hom ≫ a).app a_1) :
              (a : (shrinkFunctor.{v₁, v₁, u₁} S).toFunctor.obj a_1) → P.obj a_1)
            x :
          P.obj a_1)) ↔
    ∀ (x : S.functor.obj a_1),
      Eq (α := P.obj a_1)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((S.functorInclusion ≫ b).app a_1) :
              (a : S.functor.obj a_1) → P.obj a_1)
            x :
          P.obj a_1)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (a.app a_1) : (a : S.functor.obj a_1) → P.obj a_1) x : P.obj a_1)

Tactic:
dsimp [functor]

Premises used: CategoryTheory.Sieve.functor