-- Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology.{v, u} C
inst_1 : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
hf' : Sieve.ofArrows X' f' ∈ (J : (a : C) → Set (Sieve a)) S
D₁ D₂ : F.DescentData f
φ φ' : D₁ ⟶ D₂
i : ι
this_1 :
  Membership.mem (α := Sieve (C := Over.{v, u} (X i)) (Over.mk (𝟙 (X i)))) (γ :=
    Set (Sieve (C := Over.{v, u} (X i)) (Over.mk (𝟙 (X i)))))
    ((J.over (X i) : (a : Over (X i)) → Set (Sieve a)) (Over.mk (𝟙 (X i))))
    ((Equiv.symm (Sieve.overEquiv (Over.mk (𝟙 (X i)))) :
          (a : Sieve (Over.mk (𝟙 (X i))).left) → Sieve (Over.mk (𝟙 (X i))))
        (Sieve.pullback (f i) (Sieve.ofArrows X' f')) :
      Sieve (C := Over.{v, u} (X i)) (Over.mk (𝟙 (X i))))
Z : Over.{v, u} (X i)
g : Z ⟶ Over.mk (𝟙 (X i))
Y : C
j : ι'
p : Z.left ⟶ X' j
hp : p ≫ f' j = Over.Hom.left g ≫ f i
this_2 : Over.Hom.left g = Z.hom
this :
  ∀ (ψ : D₁ ⟶ D₂),
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((F.presheafHom (D₁.obj i) (D₂.obj i)).map g.op) :
          (a : (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))))) →
            (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op Z)))
        ((F.presheafHomObjHomEquiv :
              (a : D₁.obj i ⟶ D₂.obj i) → (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))))
            (ψ.hom i) :
          (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i))))) =
      D₁.hom (Z.hom ≫ f i) Z.hom (p ≫ p' j) ⋯ ⋯ ≫
        pullHom (Functor.map (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (ψ.hom (α j))) p (p ≫ p' j) (p ≫ p' j) ⋯ ⋯ ≫
          D₂.hom (Z.hom ≫ f i) (p ≫ p' j) Z.hom ⋯ ⋯
hφ :
  Eq (α :=
    Quiver.Hom (V := (↑(F.obj { as := op (X' j) : LocallyDiscrete Cᵒᵖ }) : Type u'))
      (Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
        (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (D₁.obj (α j)))
      (Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
        (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (D₂.obj (α j))))
    (Functor.map (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (φ.hom (α j)))
    (Functor.map (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (φ'.hom (α j)))
⊢ Eq (α := (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.presheafHom (D₁.obj i) (D₂.obj i)).map g.op) :
          (a : (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))))) →
            (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op Z)))
        ((F.presheafHomObjHomEquiv :
              (a : D₁.obj i ⟶ D₂.obj i) → (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))))
            (φ.hom i) :
          (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i))))) :
      (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.presheafHom (D₁.obj i) (D₂.obj i)).map g.op) :
          (a : (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))))) →
            (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op Z)))
        ((F.presheafHomObjHomEquiv :
              (a : D₁.obj i ⟶ D₂.obj i) → (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))))
            (φ'.hom i) :
          (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i))))) :
      (fun (X : Type v') ↦ X) ((F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op Z)))

Tactic:
simp only [this, hφ]