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⟩)
  · 

Proof state:
case h.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)))
⊢ ∀ (y : E ⟶ P.obj (op X)),
    Presieve.FamilyOfElements.IsAmalgamation x y →
      y =
        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 e he