l3lab/ntp-mathlib huggingface dataset:
/-- When `P` is a sheaf and `S` is a cover, the associated multifork is a limit. -/
def isLimitOfIsSheaf {X : C} (S : J.Cover X) (hP : IsSheaf J P) : IsLimit (S.multifork P) where
  lift := fun E : Multifork _ => hP.amalgamate S (fun I => E.ι _) fun I => E.condition _
  fac := by
    rintro (E : Multifork _) (a | b)
    · apply hP.amalgamate_map
    · rw [← E.w (WalkingMulticospan.Hom.fst b),
        ← (S.multifork P).w (WalkingMulticospan.Hom.fst b), ← assoc]
      congr 1
      apply hP.amalgamate_map
  uniq := by
    rintro (E : Multifork _) m hm
    apply hP.hom_ext S
    intro I
    erw [hm (WalkingMulticospan.left I)]
    symm
    

Proof state:
case h
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
X : C
S : GrothendieckTopology.Cover J X
hP : IsSheaf J P
E : Multifork (GrothendieckTopology.Cover.index S P)
m : E.pt ⟶ (GrothendieckTopology.Cover.multifork S P).pt
hm :
  ∀ (j : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo),
    m ≫ (GrothendieckTopology.Cover.multifork S P).π.app j = E.π.app j
I : GrothendieckTopology.Cover.Arrow S
⊢ (fun E =>
          IsSheaf.amalgamate hP S (fun I => Multifork.ι E I)
            (_ :
              ∀ (I : GrothendieckTopology.Cover.Relation S),
                Multifork.ι E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) ≫
                    MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =
                  Multifork.ι E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) ≫
                    MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))
        E ≫
      P.map I.f.op =
    E.π.app (WalkingMulticospan.left I)

Next tactic:
apply hP.amalgamate_map
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
⊢ IsSheaf J P ↔
    ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))

Next tactic:
refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
⊢ (∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))) →
    IsSheaf J P

Next tactic:
intro h E X S hS x hx
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
let T : J.Cover X := ⟨S, hS⟩
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  let T : J.Cover X := ⟨S, hS⟩
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
T : GrothendieckTopology.Cover J X := { val := S, property := hS }
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
obtain ⟨hh⟩ := h _ T
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  let T : J.Cover X := ⟨S, hS⟩
  obtain ⟨hh⟩ := h _ T
  

Proof state:
case intro
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
T : GrothendieckTopology.Cover J X := { val := S, property := hS }
hh : IsLimit (GrothendieckTopology.Cover.multifork T P)
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
let K : Multifork (T.index P) := Multifork.ofι _ E (fun I => x I.f I.hf) fun I => hx _ _ _ _ I.w
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  let T : J.Cover X := ⟨S, hS⟩
  obtain ⟨hh⟩ := h _ T
  let K : Multifork (T.index P) := Multifork.ofι _ E (fun I => x I.f I.hf) fun I => hx _ _ _ _ I.w
  

Proof state:
case intro
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
T : GrothendieckTopology.Cover J X := { val := S, property := hS }
hh : IsLimit (GrothendieckTopology.Cover.multifork T P)
K : Multifork (GrothendieckTopology.Cover.index T P) :=
  Multifork.ofι (GrothendieckTopology.Cover.index T P) E
    (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))
    (_ :
      ∀ (I : (GrothendieckTopology.Cover.index T P).R),
        (P ⋙ coyoneda.obj (op E)).map I.g₁.op
            (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =
          (P ⋙ coyoneda.obj (op E)).map I.g₂.op
            (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))
⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x t

Next tactic:
use hh.lift K
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  let T : J.Cover X := ⟨S, hS⟩
  obtain ⟨hh⟩ := h _ T
  let K : Multifork (T.index P) := Multifork.ofι _ E (fun I => x I.f I.hf) fun I => hx _ _ _ _ I.w
  use hh.lift K
  

Proof state:
case h
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
T : GrothendieckTopology.Cover J X := { val := S, property := hS }
hh : IsLimit (GrothendieckTopology.Cover.multifork T P)
K : Multifork (GrothendieckTopology.Cover.index T P) :=
  Multifork.ofι (GrothendieckTopology.Cover.index T P) E
    (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))
    (_ :
      ∀ (I : (GrothendieckTopology.Cover.index T P).R),
        (P ⋙ coyoneda.obj (op E)).map I.g₁.op
            (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =
          (P ⋙ coyoneda.obj (op E)).map I.g₂.op
            (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))
⊢ (fun t => Presieve.FamilyOfElements.IsAmalgamation x t) (IsLimit.lift hh K) ∧
    ∀ (y : (P ⋙ coyoneda.obj (op E)).obj (op X)),
      (fun t => Presieve.FamilyOfElements.IsAmalgamation x t) y → y = IsLimit.lift hh K

Next tactic:
dsimp
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  let T : J.Cover X := ⟨S, hS⟩
  obtain ⟨hh⟩ := h _ T
  let K : Multifork (T.index P) := Multifork.ofι _ E (fun I => x I.f I.hf) fun I => hx _ _ _ _ I.w
  use hh.lift K
  dsimp; constructor
  · 

Proof state:
case h.left
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
T : GrothendieckTopology.Cover J X := { val := S, property := hS }
hh : IsLimit (GrothendieckTopology.Cover.multifork T P)
K : Multifork (GrothendieckTopology.Cover.index T P) :=
  Multifork.ofι (GrothendieckTopology.Cover.index T P) E
    (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))
    (_ :
      ∀ (I : (GrothendieckTopology.Cover.index T P).R),
        (P ⋙ coyoneda.obj (op E)).map I.g₁.op
            (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =
          (P ⋙ coyoneda.obj (op E)).map I.g₂.op
            (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))
⊢ Presieve.FamilyOfElements.IsAmalgamation x
    (IsLimit.lift hh
      (Multifork.ofι (GrothendieckTopology.Cover.index { val := S, property := hS } P) E
        (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows I.f))
        (_ :
          ∀ (I : (GrothendieckTopology.Cover.index { val := S, property := hS } P).R),
            (P ⋙ coyoneda.obj (op E)).map I.g₁.op
                (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f
                  (_ :
                    (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows
                      (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f)) =
              (P ⋙ coyoneda.obj (op E)).map I.g₂.op
                (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f
                  (_ :
                    (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows
                      (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)
                          I).f)))))

Next tactic:
intro Y f hf
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  let T : J.Cover X := ⟨S, hS⟩
  obtain ⟨hh⟩ := h _ T
  let K : Multifork (T.index P) := Multifork.ofι _ E (fun I => x I.f I.hf) fun I => hx _ _ _ _ I.w
  use hh.lift K
  dsimp; constructor
  · intro Y f hf
    apply hh.fac K (WalkingMulticospan.left ⟨Y, f, hf⟩)
  · intro e he
    apply hh.uniq K
    

Proof state:
case h.right.x
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
T : GrothendieckTopology.Cover J X := { val := S, property := hS }
hh : IsLimit (GrothendieckTopology.Cover.multifork T P)
K : Multifork (GrothendieckTopology.Cover.index T P) :=
  Multifork.ofι (GrothendieckTopology.Cover.index T P) E
    (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))
    (_ :
      ∀ (I : (GrothendieckTopology.Cover.index T P).R),
        (P ⋙ coyoneda.obj (op E)).map I.g₁.op
            (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =
          (P ⋙ coyoneda.obj (op E)).map I.g₂.op
            (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))
e : E ⟶ P.obj (op X)
he : Presieve.FamilyOfElements.IsAmalgamation x e
⊢ ∀ (j : WalkingMulticospan (GrothendieckTopology.Cover.index T P).fstTo (GrothendieckTopology.Cover.index T P).sndTo),
    e ≫ (GrothendieckTopology.Cover.multifork T P).π.app j = K.π.app j

Next tactic:
rintro (a | b)
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  let T : J.Cover X := ⟨S, hS⟩
  obtain ⟨hh⟩ := h _ T
  let K : Multifork (T.index P) := Multifork.ofι _ E (fun I => x I.f I.hf) fun I => hx _ _ _ _ I.w
  use hh.lift K
  dsimp; constructor
  · intro Y f hf
    apply hh.fac K (WalkingMulticospan.left ⟨Y, f, hf⟩)
  · intro e he
    apply hh.uniq K
    rintro (a | b)
    · 

Proof state:
case h.right.x.left
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
T : GrothendieckTopology.Cover J X := { val := S, property := hS }
hh : IsLimit (GrothendieckTopology.Cover.multifork T P)
K : Multifork (GrothendieckTopology.Cover.index T P) :=
  Multifork.ofι (GrothendieckTopology.Cover.index T P) E
    (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))
    (_ :
      ∀ (I : (GrothendieckTopology.Cover.index T P).R),
        (P ⋙ coyoneda.obj (op E)).map I.g₁.op
            (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =
          (P ⋙ coyoneda.obj (op E)).map I.g₂.op
            (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))
e : E ⟶ P.obj (op X)
he : Presieve.FamilyOfElements.IsAmalgamation x e
a : (GrothendieckTopology.Cover.index T P).L
⊢ e ≫ (GrothendieckTopology.Cover.multifork T P).π.app (WalkingMulticospan.left a) = K.π.app (WalkingMulticospan.left a)

Next tactic:
apply he
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  let T : J.Cover X := ⟨S, hS⟩
  obtain ⟨hh⟩ := h _ T
  let K : Multifork (T.index P) := Multifork.ofι _ E (fun I => x I.f I.hf) fun I => hx _ _ _ _ I.w
  use hh.lift K
  dsimp; constructor
  · intro Y f hf
    apply hh.fac K (WalkingMulticospan.left ⟨Y, f, hf⟩)
  · intro e he
    apply hh.uniq K
    rintro (a | b)
    · apply he
    · 

Proof state:
case h.right.x.right
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
T : GrothendieckTopology.Cover J X := { val := S, property := hS }
hh : IsLimit (GrothendieckTopology.Cover.multifork T P)
K : Multifork (GrothendieckTopology.Cover.index T P) :=
  Multifork.ofι (GrothendieckTopology.Cover.index T P) E
    (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))
    (_ :
      ∀ (I : (GrothendieckTopology.Cover.index T P).R),
        (P ⋙ coyoneda.obj (op E)).map I.g₁.op
            (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =
          (P ⋙ coyoneda.obj (op E)).map I.g₂.op
            (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))
e : E ⟶ P.obj (op X)
he : Presieve.FamilyOfElements.IsAmalgamation x e
b : (GrothendieckTopology.Cover.index T P).R
⊢ e ≫ (GrothendieckTopology.Cover.multifork T P).π.app (WalkingMulticospan.right b) =
    K.π.app (WalkingMulticospan.right b)

Next tactic:
rw [← K.w (WalkingMulticospan.Hom.fst b), ←
        (T.multifork P).w (WalkingMulticospan.Hom.fst b), ← assoc]
theorem isSheaf_iff_multifork :
    IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P)) := by
  refine' ⟨fun hP X S => ⟨isLimitOfIsSheaf _ _ _ hP⟩, _⟩
  intro h E X S hS x hx
  let T : J.Cover X := ⟨S, hS⟩
  obtain ⟨hh⟩ := h _ T
  let K : Multifork (T.index P) := Multifork.ofι _ E (fun I => x I.f I.hf) fun I => hx _ _ _ _ I.w
  use hh.lift K
  dsimp; constructor
  · intro Y f hf
    apply hh.fac K (WalkingMulticospan.left ⟨Y, f, hf⟩)
  · intro e he
    apply hh.uniq K
    rintro (a | b)
    · apply he
    · rw [← K.w (WalkingMulticospan.Hom.fst b), ←
        (T.multifork P).w (WalkingMulticospan.Hom.fst b), ← assoc]
      

Proof state:
case h.right.x.right
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
A' : Type u₂
inst✝¹ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
P : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
h : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))
E : A
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
x : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op E)) S.arrows
hx : Presieve.FamilyOfElements.Compatible x
T : GrothendieckTopology.Cover J X := { val := S, property := hS }
hh : IsLimit (GrothendieckTopology.Cover.multifork T P)
K : Multifork (GrothendieckTopology.Cover.index T P) :=
  Multifork.ofι (GrothendieckTopology.Cover.index T P) E
    (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))
    (_ :
      ∀ (I : (GrothendieckTopology.Cover.index T P).R),
        (P ⋙ coyoneda.obj (op E)).map I.g₁.op
            (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =
          (P ⋙ coyoneda.obj (op E)).map I.g₂.op
            (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f
              (_ :
                (GrothendieckTopology.Cover.sieve T).arrows
                  (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))
e : E ⟶ P.obj (op X)
he : Presieve.FamilyOfElements.IsAmalgamation x e
b : (GrothendieckTopology.Cover.index T P).R
⊢ (e ≫
        (GrothendieckTopology.Cover.multifork T P).π.app
          (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) b))) ≫
      (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index T P)).map (WalkingMulticospan.Hom.fst b) =
    K.π.app (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) b)) ≫
      (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index T P)).map (WalkingMulticospan.Hom.fst b)

Next tactic:
congr 1