-- 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.{u_1, v₀, v, u₀, u} F J₀ J X
G : Functor.{v, v', u, u'} Cᵒᵖ A
hG₀ : Presheaf.IsSheaf J₀ (F.op ⋙ G)
hG : (X : C) → IsLimit ((data X).toOneHypercover.multifork G)
X : C
S : J.Cover X
s : Multifork (S.index G)
a : S.Arrow
i :
  (PreOneHypercover.multicospanShape (S := a.Y)
      (GrothendieckTopology.OneHypercover.toPreOneHypercover (toOneHypercover (data a.Y)))).L
X₀ : C₀
b : X₀ ⟶ PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i
Y : C
j : (data X).toPreOneHypercover.I₀
c : F.obj X₀ ⟶ (data X).toPreOneHypercover.X j
fac₁ :
  c ≫ (data X).toPreOneHypercover.f j =
    F.map b ≫ PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data a.Y)) i ≫ a.f
refine :
  ∀
    (I :
      GrothendieckTopology.Cover.Arrow
        (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} X₀) ↦ S ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀) (F.imageSieve c) ⋯)),
    Eq (α := s.pt ⟶ (F.op ⋙ G).obj (op I.Y))
      ((((lift hG₀ hG s ≫ G.map a.f.op) ≫
            Multifork.ι
              (PreOneHypercover.multifork
                (GrothendieckTopology.OneHypercover.toPreOneHypercover (toOneHypercover (data a.Y))) G)
              i) ≫
          (F.op ⋙ G).map
            (Quiver.Hom.op
              (GrothendieckTopology.Cover.Arrow.f
                { Y := X₀, f := b, hf := ⋯ :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p :=
                      fun
                        (S_1 :
                          Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)) ↦
                      Membership.mem (γ :=
                        Set (Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)))
                        ((J₀ : (a : C₀) → Set (Sieve a))
                          (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i))
                        S_1)
                      (Sieve.functorPullback F
                        (Sieve.pullback (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data a.Y)) i ≫ a.f)
                          (PreZeroHypercover.sieve₀.{u_1, v, u} (data X).toPreOneHypercover.toPreZeroHypercover)))
                      ⋯) }))) ≫
        (F.op ⋙ G).map I.f.op)
      (((s.ι a ≫
            Multifork.ι
              (PreOneHypercover.multifork
                (GrothendieckTopology.OneHypercover.toPreOneHypercover (toOneHypercover (data a.Y))) G)
              i) ≫
          (F.op ⋙ G).map
            (Quiver.Hom.op
              (GrothendieckTopology.Cover.Arrow.f
                { Y := X₀, f := b, hf := ⋯ :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p :=
                      fun
                        (S_1 :
                          Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)) ↦
                      Membership.mem (γ :=
                        Set (Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)))
                        ((J₀ : (a : C₀) → Set (Sieve a))
                          (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i))
                        S_1)
                      (Sieve.functorPullback F
                        (Sieve.pullback (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data a.Y)) i ≫ a.f)
                          (PreZeroHypercover.sieve₀.{u_1, v, u} (data X).toPreOneHypercover.toPreZeroHypercover)))
                      ⋯) }))) ≫
        (F.op ⋙ G).map I.f.op)
⊢ Eq (α :=
    s.pt ⟶
      (F.op ⋙ G).obj
        (op
          (GrothendieckTopology.Cover.Arrow.Y (J := J₀) (S :=
            Subtype.mk (p :=
              fun (S_1 : Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)) ↦
              Membership.mem (γ :=
                Set (Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)))
                ((J₀ : (a : C₀) → Set (Sieve a))
                  (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i))
                S_1)
              (Sieve.functorPullback F
                (Sieve.pullback (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data a.Y)) i ≫ a.f)
                  (PreZeroHypercover.sieve₀.{u_1, v, u} (data X).toPreOneHypercover.toPreZeroHypercover)))
              ⋯)
            { Y := X₀, f := b, hf := ⋯ })))
    (((lift hG₀ hG s ≫ G.map a.f.op) ≫
        Multifork.ι
          (PreOneHypercover.multifork
            (GrothendieckTopology.OneHypercover.toPreOneHypercover (toOneHypercover (data a.Y))) G)
          i) ≫
      (F.op ⋙ G).map
        (Quiver.Hom.op
          (GrothendieckTopology.Cover.Arrow.f
            { Y := X₀, f := b, hf := ⋯ :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p :=
                  fun (S_1 : Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)) ↦
                  Membership.mem (γ :=
                    Set (Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)))
                    ((J₀ : (a : C₀) → Set (Sieve a))
                      (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i))
                    S_1)
                  (Sieve.functorPullback F
                    (Sieve.pullback (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data a.Y)) i ≫ a.f)
                      (PreZeroHypercover.sieve₀.{u_1, v, u} (data X).toPreOneHypercover.toPreZeroHypercover)))
                  ⋯) })))
    ((s.ι a ≫
        Multifork.ι
          (PreOneHypercover.multifork
            (GrothendieckTopology.OneHypercover.toPreOneHypercover (toOneHypercover (data a.Y))) G)
          i) ≫
      (F.op ⋙ G).map
        (Quiver.Hom.op
          (GrothendieckTopology.Cover.Arrow.f
            { Y := X₀, f := b, hf := ⋯ :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p :=
                  fun (S_1 : Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)) ↦
                  Membership.mem (γ :=
                    Set (Sieve.{v₀, u₀} (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i)))
                    ((J₀ : (a : C₀) → Set (Sieve a))
                      (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data a.Y)) i))
                    S_1)
                  (Sieve.functorPullback F
                    (Sieve.pullback (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data a.Y)) i ≫ a.f)
                      (PreZeroHypercover.sieve₀.{u_1, v, u} (data X).toPreOneHypercover.toPreZeroHypercover)))
                  ⋯) })))

Tactic:
refine Presheaf.IsSheaf.hom_ext hG₀
          ⟨_, IsDenseSubsite.imageSieve_mem J₀ J F c⟩ _ _ ?_

Premises used: Subtype.mk, CategoryTheory.Presheaf.IsSheaf.hom_ext, CategoryTheory.Functor.IsDenseSubsite.imageSieve_mem