/-- An auxiliary definition to be used below.

Whenever `E` is a cone of shape `K` of sheaves, and `S` is the multifork associated to a
covering `W` of an object `X`, with respect to the cone point `E.X`, this provides a cone of
shape `K` of objects in `D`, with cone point `S.X`.

See `isLimitMultiforkOfIsLimit` for more on how this definition is used.
-/
def multiforkEvaluationCone (F : K ⥤ Sheaf J D) (E : Cone (F ⋙ sheafToPresheaf J D)) (X : C)
    (W : J.Cover X) (S : Multifork (W.index E.pt)) :
    Cone (F ⋙ sheafToPresheaf J D ⋙ (evaluation Cᵒᵖ D).obj (op X)) where
  pt := S.pt
  π :=
    { app := fun k => (Presheaf.isLimitOfIsSheaf J (F.obj k).1 W (F.obj k).2).lift <|
        Multifork.ofι _ S.pt (fun i => S.ι i ≫ (E.π.app k).app (op i.Y))
          (by
            intro i
            simp only [Category.assoc]
            erw [← (E.π.app k).naturality, ← (E.π.app k).naturality]
            dsimp
            simp only [← Category.assoc]
            congr 1
            apply S.condition)
      naturality := by
        intro i j f
        

Proof state:
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹ : Category.{max v u, w} D
K : Type z
inst✝ : Category.{z', z} K
F : K ⥤ Sheaf J D
E : Cone (F ⋙ sheafToPresheaf J D)
X : C
W : GrothendieckTopology.Cover J X
S : Multifork (GrothendieckTopology.Cover.index W E.pt)
i j : K
f : i ⟶ j
⊢ ((Functor.const K).obj S.pt).map f ≫
      (fun k =>
          IsLimit.lift (Presheaf.isLimitOfIsSheaf J (F.obj k).val W (_ : Presheaf.IsSheaf J (F.obj k).val))
            (Multifork.ofι (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt
              (fun i => Multifork.ι S i ≫ (E.π.app k).app (op i.Y))
              (_ :
                ∀ (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),
                  (fun i => Multifork.ι S i ≫ (E.π.app k).app (op i.Y))
                        (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) ≫
                      MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =
                    (fun i => Multifork.ι S i ≫ (E.π.app k).app (op i.Y))
                        (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) ≫
                      MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i)))
        j =
    (fun k =>
          IsLimit.lift (Presheaf.isLimitOfIsSheaf J (F.obj k).val W (_ : Presheaf.IsSheaf J (F.obj k).val))
            (Multifork.ofι (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt
              (fun i => Multifork.ι S i ≫ (E.π.app k).app (op i.Y))
              (_ :
                ∀ (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),
                  (fun i => Multifork.ι S i ≫ (E.π.app k).app (op i.Y))
                        (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) ≫
                      MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =
                    (fun i => Multifork.ι S i ≫ (E.π.app k).app (op i.Y))
                        (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) ≫
                      MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i)))
        i ≫
      (F ⋙ sheafToPresheaf J D ⋙ (evaluation Cᵒᵖ D).obj (op X)).map f

Next tactic:
dsimp [Presheaf.isLimitOfIsSheaf]