-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X : C
X₀ : C₀
f : F.obj X₀ ⟶ X
Z₁ : C₀
g₁ : Z₁ ⟶ X₀
h₁ : (data X).SieveStruct f g₁
Z₂ : C₀
g₂ : Z₂ ⟶ X₀
h₂ : (data X).SieveStruct f g₂
T₀ : C₀
p₁ :
  T₀ ⟶
    GrothendieckTopology.Cover.Arrow.Y (J := J₀) (S :=
      Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f) ⋯)
      { Y := Z₁, f := g₁, hf := ⋯ }
p₂ :
  T₀ ⟶
    GrothendieckTopology.Cover.Arrow.Y (J := J₀) (S :=
      Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f) ⋯)
      { Y := Z₂, f := g₂, hf := ⋯ }
w :
  p₁ ≫
      GrothendieckTopology.Cover.Arrow.f
        { Y := Z₁, f := g₁, hf := ⋯ :
          GrothendieckTopology.Cover.Arrow
            (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) ((data X).sieve f)
              ⋯) } =
    p₂ ≫ GrothendieckTopology.Cover.Arrow.f { Y := Z₂, f := g₂, hf := ⋯ }
dsimp :
  Eq (α := presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op T₀))
    (restriction.res data G₀ (X₀ := X₀) (f := f) (g := g₁) (Nonempty.some ⋯) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map p₁.op)
    (restriction.res data G₀ (X₀ := X₀) (f := f) (g := g₂) (Nonempty.some ⋯) ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map p₂.op)
⊢ Eq (α :=
    presheafObj data G₀ X ⟶
      G₀.obj (P := Presheaf.IsSheaf J₀).obj
        (op
          { Z := T₀, g₁ := p₁, g₂ := p₂, w := w :
              GrothendieckTopology.Cover.Arrow.Relation (J := J₀) (S :=
                Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                  ((data X).sieve f) ⋯)
                { Y := Z₁, f := g₁, hf := ⋯ :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                      ((data X).sieve f) ⋯) }
                { Y := Z₂, f := g₂, hf := ⋯ :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                      ((data X).sieve f) ⋯) } }.Z))
    ((fun
            (x :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                  ((data X).sieve f) ⋯)) ↦
          (match (motive :=
              (x :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                      ((data X).sieve f) ⋯)) →
                presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op x.Y))
              x with
            | { Y := Y₀, f := g, hf := hg } => restriction.res data G₀ (Nonempty.some hg) :
            presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op x.Y)))
        { Y := Z₁, f := g₁, hf := ⋯ } ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := w :
              GrothendieckTopology.Cover.Arrow.Relation (J := J₀) (S :=
                Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                  ((data X).sieve f) ⋯)
                { Y := Z₁, f := g₁, hf := ⋯ :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                      ((data X).sieve f) ⋯) }
                { Y := Z₂, f := g₂, hf := ⋯ :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                      ((data X).sieve f) ⋯) } }.g₁.op)
    ((fun
            (x :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                  ((data X).sieve f) ⋯)) ↦
          (match (motive :=
              (x :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                      ((data X).sieve f) ⋯)) →
                presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op x.Y))
              x with
            | { Y := Y₀, f := g, hf := hg } => restriction.res data G₀ (Nonempty.some hg) :
            presheafObj data G₀ X ⟶ G₀.obj (P := Presheaf.IsSheaf J₀).obj (op x.Y)))
        { Y := Z₂, f := g₂, hf := ⋯ } ≫
      G₀.obj (P := Presheaf.IsSheaf J₀).map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := w :
              GrothendieckTopology.Cover.Arrow.Relation (J := J₀) (S :=
                Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                  ((data X).sieve f) ⋯)
                { Y := Z₁, f := g₁, hf := ⋯ :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                      ((data X).sieve f) ⋯) }
                { Y := Z₂, f := g₂, hf := ⋯ :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀)
                      ((data X).sieve f) ⋯) } }.g₂.op)

Tactic:
dsimp at g₁ g₂ p₁ p₂ w ⊢