awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
X : C
W : J.Cover X
S : Multifork (W.index E.pt)
m : S.pt ⟶ (W.multifork E.pt).pt
hm : ∀ (i : W.shape.L), m ≫ (W.multifork E.pt).ι i = S.ι i
⊢ m =
    (fun (S : Multifork (W.index E.pt)) ↦
        (IsLimit.lift (isLimitOfPreserves (Functor.obj (evaluation Cᵒᵖ D) (op X)) hE)
            (multiforkEvaluationCone F E X W S) :
          (multiforkEvaluationCone F E X W S).pt ⟶
            Cone.pt (F := (F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
              (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E)))
      S
have apply : ∀ (j : K),
  Eq (α := S.pt ⟶ ((F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)).obj j)
    (m ≫ NatTrans.app (Cone.π (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E)) j)
    (CategoryStruct.comp (Y :=
      Cone.pt (F := (F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
        (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E))
      ((fun (S : Multifork (W.index E.pt)) ↦
            (IsLimit.lift (isLimitOfPreserves (Functor.obj (evaluation Cᵒᵖ D) (op X)) hE)
                (multiforkEvaluationCone F E X W S) :
              (multiforkEvaluationCone F E X W S).pt ⟶
                Cone.pt (F := (F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
                  (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E)))
          S :
        (multiforkEvaluationCone F E X W S).pt ⟶
          Cone.pt (F := (F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
            (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E))
      (NatTrans.app (Cone.π (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E)) j :
        Functor.obj.{z', w', z, w}
            (Functor.obj (Functor.const K)
              (Cone.pt (F := (F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
                (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E)))
            j ⟶
          ((F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)).obj j))
Tactic: apply (isLimitOfPreserves ((evaluation Cᵒᵖ D).obj (op X)) hE).hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
X : C
W : J.Cover X
S : Multifork (W.index E.pt)
m : S.pt ⟶ (W.multifork E.pt).pt
hm : ∀ (i : W.shape.L), m ≫ (W.multifork E.pt).ι i = S.ι i
k : K
⊢ Eq (α := S.pt ⟶ ((F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)).obj k)
    (m ≫ NatTrans.app (Cone.π (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E)) k)
    (CategoryStruct.comp (Y :=
      Cone.pt (F := (F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
        (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E))
      ((fun (S : Multifork (W.index E.pt)) ↦
            (IsLimit.lift (isLimitOfPreserves (Functor.obj (evaluation Cᵒᵖ D) (op X)) hE)
                (multiforkEvaluationCone F E X W S) :
              (multiforkEvaluationCone F E X W S).pt ⟶
                Cone.pt (F := (F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
                  (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E)))
          S :
        (multiforkEvaluationCone F E X W S).pt ⟶
          Cone.pt (F := (F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
            (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E))
      (NatTrans.app (Cone.π (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E)) k :
        Functor.obj.{z', w', z, w}
            (Functor.obj (Functor.const K)
              (Cone.pt (F := (F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
                (Functor.mapCone (Functor.obj (evaluation Cᵒᵖ D) (op X)) E)))
            k ⟶
          ((F ⋙ sheafToPresheaf J D) ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)).obj k))
have dsimp : m ≫ (E.π.app k).app (op X) =
  IsLimit.lift (isLimitOfPreserves (Functor.obj (evaluation Cᵒᵖ D) (op X)) hE) (multiforkEvaluationCone F E X W S) ≫
    (E.π.app k).app (op X)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
X : C
W : J.Cover X
S : Multifork (W.index E.pt)
m : S.pt ⟶ (W.multifork E.pt).pt
hm : ∀ (i : W.shape.L), m ≫ (W.multifork E.pt).ι i = S.ι i
k : K
⊢ m ≫ (E.π.app k).app (op X) =
    IsLimit.lift (isLimitOfPreserves (Functor.obj (evaluation Cᵒᵖ D) (op X)) hE) (multiforkEvaluationCone F E X W S) ≫
      (E.π.app k).app (op X)
have erw : m ≫ (E.π.app k).app (op X) = (multiforkEvaluationCone F E X W S).π.app k
Tactic: erw [(isLimitOfPreserves ((evaluation Cᵒᵖ D).obj (op X)) hE).fac]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
X : C
W : J.Cover X
S : Multifork (W.index E.pt)
m : S.pt ⟶ (W.multifork E.pt).pt
hm : ∀ (i : W.shape.L), m ≫ (W.multifork E.pt).ι i = S.ι i
k : K
⊢ m ≫ (E.π.app k).app (op X) = (multiforkEvaluationCone F E X W S).π.app k
have h : ∀ (I : W.Arrow),
  (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map I.f.op =
    (multiforkEvaluationCone F E X W S).π.app k ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map I.f.op
Tactic: apply Presheaf.IsSheaf.hom_ext (F.obj k).2 W
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
X : C
W : J.Cover X
S : Multifork (W.index E.pt)
m : S.pt ⟶ (W.multifork E.pt).pt
hm : ∀ (i : W.shape.L), m ≫ (W.multifork E.pt).ι i = S.ι i
k : K
i : W.Arrow
⊢ (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op =
    (multiforkEvaluationCone F E X W S).π.app k ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op
have h : (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op =
  sorry.amalgamate W
      (fun (x : W.Arrow) ↦
        (Multifork.ofι (W.index ((F.obj k).obj (P := Presheaf.IsSheaf J))) S.pt
              (fun (i : W.shape.L) ↦ S.ι i ≫ (E.π.app k).app (op i.Y)) sorry).ι
          x)
      sorry ≫
    (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op
Tactic: dsimp only [multiforkEvaluationCone, Presheaf.isLimitOfIsSheaf]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
X : C
W : J.Cover X
S : Multifork (W.index E.pt)
m : S.pt ⟶ (W.multifork E.pt).pt
hm : ∀ (i : W.shape.L), m ≫ (W.multifork E.pt).ι i = S.ι i
k : K
i : W.Arrow
⊢ (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op =
    ⋯.amalgamate W
        (fun (x : W.Arrow) ↦
          (Multifork.ofι (W.index ((F.obj k).obj (P := Presheaf.IsSheaf J))) S.pt
                (fun (i : W.shape.L) ↦ S.ι i ≫ (E.π.app k).app (op i.Y)) ⋯).ι
            x)
        ⋯ ≫
      (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op
have h : (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op =
  (Multifork.ofι (W.index ((F.obj k).obj (P := Presheaf.IsSheaf J))) S.pt
        (fun (i : W.shape.L) ↦ S.ι i ≫ (E.π.app k).app (op i.Y)) sorry).ι
    i
Tactic: rw [(F.obj k).property.amalgamate_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
X : C
W : J.Cover X
S : Multifork (W.index E.pt)
m : S.pt ⟶ (W.multifork E.pt).pt
hm : ∀ (i : W.shape.L), m ≫ (W.multifork E.pt).ι i = S.ι i
k : K
i : W.Arrow
⊢ (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op =
    (Multifork.ofι (W.index ((F.obj k).obj (P := Presheaf.IsSheaf J))) S.pt
          (fun (i : W.shape.L) ↦ S.ι i ≫ (E.π.app k).app (op i.Y)) ⋯).ι
      i
have h : (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op =
  Multifork.ι
    { pt := S.pt,
      π :=
        {
          app := fun (x : WalkingMulticospan W.shape) ↦
            match (motive :=
              (x : WalkingMulticospan W.shape) →
                S.pt ⟶
                  match (motive := WalkingMulticospan W.shape → D) x with
                  | WalkingMulticospan.left a => (F.obj k).obj (P := Presheaf.IsSheaf J).obj (op a.Y)
                  | WalkingMulticospan.right b => (F.obj k).obj (P := Presheaf.IsSheaf J).obj (op b.r.Z))
              x with
            | WalkingMulticospan.left a => S.ι a ≫ (E.π.app k).app (op a.Y)
            | WalkingMulticospan.right b =>
              (S.ι b.fst ≫ (E.π.app k).app (op b.fst.Y)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map b.r.g₁.op,
          naturality := sorry } }
    i
Tactic: dsimp [Multifork.ofι]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
X : C
W : J.Cover X
S : Multifork (W.index E.pt)
m : S.pt ⟶ (W.multifork E.pt).pt
hm : ∀ (i : W.shape.L), m ≫ (W.multifork E.pt).ι i = S.ι i
k : K
i : W.Arrow
⊢ (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op =
    Multifork.ι
      { pt := S.pt,
        π :=
          {
            app := fun (x : WalkingMulticospan W.shape) ↦
              match (motive :=
                (x : WalkingMulticospan W.shape) →
                  S.pt ⟶
                    match (motive := WalkingMulticospan W.shape → D) x with
                    | WalkingMulticospan.left a => (F.obj k).obj (P := Presheaf.IsSheaf J).obj (op a.Y)
                    | WalkingMulticospan.right b => (F.obj k).obj (P := Presheaf.IsSheaf J).obj (op b.r.Z))
                x with
              | WalkingMulticospan.left a => S.ι a ≫ (E.π.app k).app (op a.Y)
              | WalkingMulticospan.right b =>
                (S.ι b.fst ≫ (E.π.app k).app (op b.fst.Y)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map b.r.g₁.op,
            naturality := ⋯ } }
      i
have h : (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op = S.ι i ≫ (E.π.app k).app (op i.Y)
Tactic: change _ = S.ι i ≫ _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
X : C
W : J.Cover X
S : Multifork (W.index E.pt)
m : S.pt ⟶ (W.multifork E.pt).pt
hm : ∀ (i : W.shape.L), m ≫ (W.multifork E.pt).ι i = S.ι i
k : K
i : W.Arrow
⊢ (m ≫ (E.π.app k).app (op X)) ≫ (F.obj k).obj (P := Presheaf.IsSheaf J).map i.f.op = S.ι i ≫ (E.π.app k).app (op i.Y)
have h : m ≫ (Functor.obj (Functor.obj (Functor.const K) E.pt) k).map i.f.op ≫ (E.π.app k).app (op i.Y) =
  m ≫ (W.multifork E.pt).ι i ≫ (E.π.app k).app (op i.Y)
Tactic: erw [← hm, Category.assoc, ← (E.π.app k).naturality, Category.assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasLimitsOfShape.{z', z, w', w} K D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cone (F ⋙ sheafToPresheaf J D)
hE : IsLimit E
⊢ Presheaf.IsSheaf J E.pt
have rw : ∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork E.pt))
Tactic: rw [Presheaf.isSheaf_iff_multifork]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasWeakSheafify.{v, w', u, w} J D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cocone (F ⋙ sheafToPresheaf J D)
k : K
⊢ ((sheafifyCocone E).ι.app k).hom = E.ι.app k ≫ toSheafify J E.pt
have rw : Eq (α :=
  Functor.obj (sheafToPresheaf J D) (Functor.obj (sheafToPresheaf J D ⋙ presheafToSheaf J D) (F.obj k)) ⟶
    (Functor.obj (Functor.obj (Functor.const K) (sheafifyCocone E).pt) k).obj (P := Presheaf.IsSheaf J))
  (Functor.map (sheafToPresheaf J D) (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
    ((sheafifyCocone E).ι.app k).hom)
  (Functor.map (sheafToPresheaf J D) (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
    E.ι.app k ≫ toSheafify J E.pt)
Tactic: rw [← cancel_epi ((sheafToPresheaf _ _).map
    ((sheafificationAdjunction J D).counit.app (F.obj k)))]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasWeakSheafify.{v, w', u, w} J D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cocone (F ⋙ sheafToPresheaf J D)
k : K
⊢ Eq (α :=
    Functor.obj (sheafToPresheaf J D) (Functor.obj (sheafToPresheaf J D ⋙ presheafToSheaf J D) (F.obj k)) ⟶
      (Functor.obj (Functor.obj (Functor.const K) (sheafifyCocone E).pt) k).obj (P := Presheaf.IsSheaf J))
    (Functor.map (sheafToPresheaf J D) (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      ((sheafifyCocone E).ι.app k).hom)
    (Functor.map (sheafToPresheaf J D) (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      E.ι.app k ≫ toSheafify J E.pt)
have dsimp : Eq (α :=
  (Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J) ⟶
    (Functor.obj (presheafToSheaf J D) E.pt).obj (P := Presheaf.IsSheaf J))
  (InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
    InducedCategory.Hom.hom (NatTrans.app (inv (Adjunction.counit (sheafificationAdjunction J D))) (F.obj k)) ≫
      InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)))
  (InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
    E.ι.app k ≫ toSheafify J E.pt)
Tactic: dsimp [sheafifyCocone]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasWeakSheafify.{v, w', u, w} J D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cocone (F ⋙ sheafToPresheaf J D)
k : K
⊢ Eq (α :=
    (Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J) ⟶
      (Functor.obj (presheafToSheaf J D) E.pt).obj (P := Presheaf.IsSheaf J))
    (InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      InducedCategory.Hom.hom (NatTrans.app (inv (Adjunction.counit (sheafificationAdjunction J D))) (F.obj k)) ≫
        InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)))
    (InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      E.ι.app k ≫ toSheafify J E.pt)
have rw : InducedCategory.Hom.hom (𝟙 (Functor.obj (sheafToPresheaf J D ⋙ presheafToSheaf J D) (F.obj k))) ≫
    InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)) =
  InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
    E.ι.app k ≫ toSheafify J E.pt
Tactic: rw [← ObjectProperty.FullSubcategory.comp_hom_assoc,
    ← NatTrans.comp_app, IsIso.hom_inv_id, NatTrans.id_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasWeakSheafify.{v, w', u, w} J D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cocone (F ⋙ sheafToPresheaf J D)
k : K
⊢ InducedCategory.Hom.hom (𝟙 (Functor.obj (sheafToPresheaf J D ⋙ presheafToSheaf J D) (F.obj k))) ≫
      InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)) =
    InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      E.ι.app k ≫ toSheafify J E.pt
have dsimp : 𝟙 ((Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J)) ≫
    InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)) =
  InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
    E.ι.app k ≫ toSheafify J E.pt
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasWeakSheafify.{v, w', u, w} J D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cocone (F ⋙ sheafToPresheaf J D)
k : K
⊢ 𝟙 ((Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J)) ≫
      InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)) =
    InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      E.ι.app k ≫ toSheafify J E.pt
have rw : Eq (α :=
  (Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J) ⟶
    (Functor.obj (presheafToSheaf J D) E.pt).obj (P := Presheaf.IsSheaf J))
  (InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)))
  (InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
    E.ι.app k ≫ toSheafify J E.pt)
Tactic: rw [Category.id_comp, toSheafify_naturality, sheafificationAdjunction_counit_app_val,
    sheafifyLift_id_toSheafify_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasWeakSheafify.{v, w', u, w} J D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cocone (F ⋙ sheafToPresheaf J D)
k : K
rw :
  Eq (α :=
    (Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J) ⟶
      (Functor.obj (presheafToSheaf J D) E.pt).obj (P := Presheaf.IsSheaf J))
    (InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)))
    (InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      E.ι.app k ≫ toSheafify J E.pt)
⊢ 𝟙 ((Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J)) ≫
      InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)) =
    InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      E.ι.app k ≫ toSheafify J E.pt
have rw₁ : Eq (α :=
  (Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J) ⟶
    (Functor.obj (presheafToSheaf J D) E.pt).obj (P := Presheaf.IsSheaf J))
  (InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)))
  (InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
    toSheafify J ((F.obj k).obj (P := Presheaf.IsSheaf J)) ≫ sheafifyMap J (E.ι.app k))
Tactic: rw [Category.id_comp, toSheafify_naturality, sheafificationAdjunction_counit_app_val,
    sheafifyLift_id_toSheafify_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Limits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
K : Type z
inst_2 : Category.{z', z} K
inst_3 : HasWeakSheafify.{v, w', u, w} J D
F : K ⥤ Sheaf.{v, w', u, w} J D
E : Cocone (F ⋙ sheafToPresheaf J D)
k : K
rw :
  Eq (α :=
    (Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J) ⟶
      (Functor.obj (presheafToSheaf J D) E.pt).obj (P := Presheaf.IsSheaf J))
    (InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)))
    (InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      E.ι.app k ≫ toSheafify J E.pt)
rw₁ :
  Eq (α :=
    (Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J) ⟶
      (Functor.obj (presheafToSheaf J D) E.pt).obj (P := Presheaf.IsSheaf J))
    (InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)))
    (InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      toSheafify J ((F.obj k).obj (P := Presheaf.IsSheaf J)) ≫ sheafifyMap J (E.ι.app k))
⊢ 𝟙 ((Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J)) ≫
      InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)) =
    InducedCategory.Hom.hom (NatTrans.app (Adjunction.counit (sheafificationAdjunction J D)) (F.obj k)) ≫
      E.ι.app k ≫ toSheafify J E.pt
have rw₂ : Eq (α :=
  (Functor.obj (presheafToSheaf J D) ((F.obj k).obj (P := Presheaf.IsSheaf J))).obj (P := Presheaf.IsSheaf J) ⟶
    (Functor.obj (presheafToSheaf J D) E.pt).obj (P := Presheaf.IsSheaf J))
  (InducedCategory.Hom.hom (Functor.map (presheafToSheaf J D) (E.ι.app k)))
  (sheafifyLift J (𝟙 ((F.obj k).obj (P := Presheaf.IsSheaf J))) sorry ≫
    toSheafify J ((F.obj k).obj (P := Presheaf.IsSheaf J)) ≫ sheafifyMap J (E.ι.app k))
Tactic: rw [Category.id_comp, toSheafify_naturality, sheafificationAdjunction_counit_app_val,
    sheafifyLift_id_toSheafify_assoc]