-- Module: Mathlib.CategoryTheory.Topos.Sheaf
Goal:
C : Type u
inst : Category.{v, u} C
F G : Cᵒᵖ ⥤ Type (max u v)
m : F ⟶ G
χ' : G ⟶ Functor.sieves C
X : Cᵒᵖ
x : G.obj X
h₁ : ∀ (x : Cᵒᵖ), m.app x ≫ χ'.app x = Types.isTerminalPUnit.from (F.obj x) ≫ ↾fun (x_1 : PUnit.{(max u v) + 1}) ↦ ⊤
h₂ :
  ∀ (x : Cᵒᵖ) (x₁ y₁ : F.obj x),
    Eq (α := G.obj x) ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app x) : (a : F.obj x) → G.obj x) x₁ : G.obj x)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app x) : (a : F.obj x) → G.obj x) y₁ : G.obj x) →
      x₁ = y₁
h₃ :
  ∀ (x : Cᵒᵖ) (x₂ : G.obj x),
    Eq (α := Sieve.{v, u} (Opposite.unop x))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (χ'.app x) : (a : G.obj x) → Sieve (Opposite.unop x)) x₂ :
          Sieve.{v, u} (Opposite.unop x))
        (⊤ : Sieve.{v, u} (Opposite.unop x)) →
      ∃ (x₁ : F.obj x), (ConcreteCategory.hom (FC := TypeCat.Fun) (m.app x) : (a : F.obj x) → G.obj x) x₁ = x₂
Y : C
f : Y ⟶ Opposite.unop X
this :
  Eq (α := Functor.obj (Functor.sieves C) (Opposite.op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map (Functor.sieves C) f.op) :
          (a : (Functor.sieves C).obj (Opposite.op (Opposite.unop X))) → (Functor.sieves C).obj (Opposite.op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (χ'.app X) :
            (a : (fun (X : Type (max u v)) ↦ X) (G.obj X)) → (fun (X : Type (max u v)) ↦ X) ((Functor.sieves C).obj X))
          x) :
      Functor.obj (Functor.sieves C) (Opposite.op Y))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (↾fun (S : Sieve.{v, u} (Opposite.unop (Opposite.op (Opposite.unop X)))) ↦ Sieve.pullback f.op.unop S) :
          (a : (Functor.sieves C).obj (Opposite.op (Opposite.unop X))) → (Functor.sieves C).obj (Opposite.op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (χ'.app X) :
            (a : (fun (X : Type (max u v)) ↦ X) (G.obj X)) → (fun (X : Type (max u v)) ↦ X) ((Functor.sieves C).obj X))
          x) :
      Functor.obj (Functor.sieves C) (Opposite.op Y))
w :
  Eq (α := Sieve.{v, u} Y)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (χ'.app (Opposite.op Y)) : (a : G.obj (Opposite.op Y)) → Sieve Y)
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f.op) : (a : G.obj X) → G.obj (Opposite.op Y)) x) :
        Sieve.{v, u} Y)
      (⊤ : Sieve.{v, u} Y) ↔
    ∃ (a : F.obj (Opposite.op Y)),
      Eq (α := G.obj (Opposite.op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f.op) : (a : G.obj X) → G.obj (Opposite.op Y)) x :
          G.obj (Opposite.op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app (Opposite.op Y)) :
              (a : F.obj (Opposite.op Y)) → G.obj (Opposite.op Y))
            a :
          G.obj (Opposite.op Y))
⊢ Sieve.pullback f ((ConcreteCategory.hom (FC := TypeCat.Fun) (χ'.app X) : (a : G.obj X) → Sieve (Opposite.unop X)) x) =
      ⊤ ↔
    ∃ (a : F.obj (Opposite.op Y)),
      Eq (α := G.obj (Opposite.op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map f.op) : (a : G.obj X) → G.obj (Opposite.op Y)) x :
          G.obj (Opposite.op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (m.app (Opposite.op Y)) :
              (a : F.obj (Opposite.op Y)) → G.obj (Opposite.op Y))
            a :
          G.obj (Opposite.op Y))

Tactic:
rw [← dsimp% this, ← dsimp% NatTrans.naturality_apply χ' f.op x]

Premises used: CategoryTheory.NatTrans.naturality_apply, Quiver.Hom.op