-- Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage
Goal:
C : Type u
inst : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
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
D₁ D₂ : F.DescentData f
φ :
  Quiver.Hom (V := F.DescentData f') (Functor.obj (pullFunctor F (p := 𝟙 S) (α := α) (p' := p') ⋯) D₁)
    (Functor.obj (pullFunctor F (p := 𝟙 S) (α := α) (p' := p') ⋯) D₂)
i : ι
Z : C
q : Z ⟶ X i
j₁ : ι'
a₁ : Z ⟶ X' j₁
j₂ : ι'
a₂ : Z ⟶ X' j₂
fac₁ : a₁ ≫ f' j₁ = q ≫ f i
fac₂ : a₂ ≫ f' j₂ = q ≫ f i
this :
  ((F.mapComp' (p' j₁).op.toLoc a₁.op.toLoc
                  (CategoryStruct.comp (Y := { as := op (X' j₁) : LocallyDiscrete Cᵒᵖ }) (p' j₁).op.toLoc a₁.op.toLoc)
                  ⋯).inv.toNatTrans.app
          (D₁.obj (α j₁)) ≫
        pullHom (φ.hom j₁) a₁ (a₁ ≫ p' j₁) (a₁ ≫ p' j₁) ⋯ ⋯ ≫
          (F.mapComp' (p' j₁).op.toLoc a₁.op.toLoc
                    (CategoryStruct.comp (Y := { as := op (X' j₁) : LocallyDiscrete Cᵒᵖ }) (p' j₁).op.toLoc a₁.op.toLoc)
                    ⋯).hom.toNatTrans.app
            (D₂.obj (α j₁))) ≫
      (F.mapComp' (p' j₁).op.toLoc a₁.op.toLoc
                  (CategoryStruct.comp (Y := { as := op (X' j₁) : LocallyDiscrete Cᵒᵖ }) (p' j₁).op.toLoc a₁.op.toLoc)
                  ⋯).inv.toNatTrans.app
          (D₂.obj (α j₁)) ≫
        D₂.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫
          (F.mapComp' (p' j₂).op.toLoc a₂.op.toLoc
                    (CategoryStruct.comp (Y := { as := op (X' j₂) : LocallyDiscrete Cᵒᵖ }) (p' j₂).op.toLoc a₂.op.toLoc)
                    ⋯).hom.toNatTrans.app
            (D₂.obj (α j₂)) =
    ((F.mapComp' (p' j₁).op.toLoc a₁.op.toLoc
                  (CategoryStruct.comp (Y := { as := op (X' j₁) : LocallyDiscrete Cᵒᵖ }) (p' j₁).op.toLoc a₁.op.toLoc)
                  ⋯).inv.toNatTrans.app
          (D₁.obj (α j₁)) ≫
        D₁.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫
          (F.mapComp' (p' j₂).op.toLoc a₂.op.toLoc
                    (CategoryStruct.comp (Y := { as := op (X' j₂) : LocallyDiscrete Cᵒᵖ }) (p' j₂).op.toLoc a₂.op.toLoc)
                    ⋯).hom.toNatTrans.app
            (D₁.obj (α j₂))) ≫
      (F.mapComp' (p' j₂).op.toLoc a₂.op.toLoc
                  (CategoryStruct.comp (Y := { as := op (X' j₂) }) (p' j₂).op.toLoc a₂.op.toLoc) ⋯).inv.toNatTrans.app
          (D₁.obj (α j₂)) ≫
        pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯ ≫
          (F.mapComp' (p' j₂).op.toLoc a₂.op.toLoc
                    (CategoryStruct.comp (Y := { as := op (X' j₂) : LocallyDiscrete Cᵒᵖ }) (p' j₂).op.toLoc a₂.op.toLoc)
                    ⋯).hom.toNatTrans.app
            (D₂.obj (α j₂))
⊢ pullHom (φ.hom j₁) a₁ (a₁ ≫ p' j₁) (a₁ ≫ p' j₁) ⋯ ⋯ ≫
      D₂.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫
        (F.mapComp' (p' j₂).op.toLoc a₂.op.toLoc
                  (CategoryStruct.comp (Y := { as := op (X' j₂) : LocallyDiscrete Cᵒᵖ }) (p' j₂).op.toLoc a₂.op.toLoc)
                  ⋯).hom.toNatTrans.app
          (D₂.obj (α j₂)) =
    D₁.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫
      pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯ ≫
        (F.mapComp' (p' j₂).op.toLoc a₂.op.toLoc
                  (CategoryStruct.comp (Y := { as := op (X' j₂) }) (p' j₂).op.toLoc a₂.op.toLoc) ⋯).hom.toNatTrans.app
          (D₂.obj (α j₂))

Tactic:
simp only [Category.assoc, Cat.Hom.hom_inv_id_toNatTrans_app_assoc, cancel_epi] at this

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Cat.Hom.hom_inv_id_toNatTrans_app_assoc, CategoryTheory.cancel_epi