awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u_1
B : Type u_2
inst_1 : Category.{v_1, u_1} A
inst_2 : Category.{v_2, u_2} B
F : Functor.{v_1, v_2, u_1, u_2} A B
G₁ : Functor.{v, v_1, u, u_1} Cᵒᵖ A ⥤ Sheaf.{v, v_1, u, u_1} J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : Functor.{v, v_2, u, u_2} Cᵒᵖ B ⥤ Sheaf.{v, v_2, u, u_2} J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst_3 : J.HasSheafCompose F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ A
h :
  J.W (whiskerRight (adj₁.unit.app P) F) ↔
    IsIso (Y := (G₁ ⋙ sheafCompose J F).obj P) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P)
⊢ IsIso (G₂.map (whiskerRight (adj₁.unit.app P) F)) ↔
    IsIso (Y := (G₁ ⋙ sheafCompose J F).obj P) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P)
have h₁ : J.W (whiskerRight (adj₁.unit.app P) F) ↔
  J.W (X := obj (sheafToPresheaf J B) ((obj (whiskeringRight Cᵒᵖ A B) F ⋙ G₂).obj P)) (Y :=
    obj (sheafToPresheaf J B) ((G₁ ⋙ sheafCompose J F).obj P))
    (map (sheafToPresheaf J B) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P))
Tactic: rw [← J.W_iff_isIso_map_of_adjunction adj₂, ← J.W_sheafToPresheaf_map_iff_isIso,
    ← sheafComposeNatTrans_fac J F adj₁ adj₂,
    (W _).precomp_iff _ _ (J.W_adj_unit_app adj₂ (P ⋙ F))]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u_1
B : Type u_2
inst_1 : Category.{v_1, u_1} A
inst_2 : Category.{v_2, u_2} B
F : Functor.{v_1, v_2, u_1, u_2} A B
G₁ : Functor.{v, v_1, u, u_1} Cᵒᵖ A ⥤ Sheaf.{v, v_1, u, u_1} J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : Functor.{v, v_2, u, u_2} Cᵒᵖ B ⥤ Sheaf.{v, v_2, u, u_2} J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst_3 : J.HasSheafCompose F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ A
h :
  J.W (whiskerRight (adj₁.unit.app P) F) ↔
    IsIso (Y := (G₁ ⋙ sheafCompose J F).obj P) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P)
h₁ :
  J.W (whiskerRight (adj₁.unit.app P) F) ↔
    J.W (X := obj (sheafToPresheaf J B) ((obj (whiskeringRight Cᵒᵖ A B) F ⋙ G₂).obj P)) (Y :=
      obj (sheafToPresheaf J B) ((G₁ ⋙ sheafCompose J F).obj P))
      (map (sheafToPresheaf J B) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P))
⊢ IsIso (G₂.map (whiskerRight (adj₁.unit.app P) F)) ↔
    IsIso (Y := (G₁ ⋙ sheafCompose J F).obj P) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P)
have h₂ : J.W (X := obj (𝟭 (Functor.{v, v_1, u, u_1} Cᵒᵖ A)) P ⋙ F) (Y := (G₁ ⋙ sheafToPresheaf J A).obj P ⋙ F)
    (adj₂.unit.app (P ⋙ F) ≫ map (sheafToPresheaf J B) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P)) ↔
  J.W (X := obj (sheafToPresheaf J B) ((obj (whiskeringRight Cᵒᵖ A B) F ⋙ G₂).obj P)) (Y :=
    obj (sheafToPresheaf J B) ((G₁ ⋙ sheafCompose J F).obj P))
    (map (sheafToPresheaf J B) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P))
Tactic: rw [← J.W_iff_isIso_map_of_adjunction adj₂, ← J.W_sheafToPresheaf_map_iff_isIso,
    ← sheafComposeNatTrans_fac J F adj₁ adj₂,
    (W _).precomp_iff _ _ (J.W_adj_unit_app adj₂ (P ⋙ F))]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u_1
B : Type u_2
inst_1 : Category.{v_1, u_1} A
inst_2 : Category.{v_2, u_2} B
F : Functor.{v_1, v_2, u_1, u_2} A B
G₁ : Functor.{v, v_1, u, u_1} Cᵒᵖ A ⥤ Sheaf.{v, v_1, u, u_1} J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : Functor.{v, v_2, u, u_2} Cᵒᵖ B ⥤ Sheaf.{v, v_2, u, u_2} J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst_3 : J.HasSheafCompose F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ A
h :
  J.W (whiskerRight (adj₁.unit.app P) F) ↔
    IsIso (Y := (G₁ ⋙ sheafCompose J F).obj P) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P)
h₁ :
  J.W (whiskerRight (adj₁.unit.app P) F) ↔
    J.W (X := obj (sheafToPresheaf J B) ((obj (whiskeringRight Cᵒᵖ A B) F ⋙ G₂).obj P)) (Y :=
      obj (sheafToPresheaf J B) ((G₁ ⋙ sheafCompose J F).obj P))
      (map (sheafToPresheaf J B) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P))
h₂ :
  J.W (X := obj (𝟭 (Functor.{v, v_1, u, u_1} Cᵒᵖ A)) P ⋙ F) (Y := (G₁ ⋙ sheafToPresheaf J A).obj P ⋙ F)
      (adj₂.unit.app (P ⋙ F) ≫ map (sheafToPresheaf J B) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P)) ↔
    J.W (X := obj (sheafToPresheaf J B) ((obj (whiskeringRight Cᵒᵖ A B) F ⋙ G₂).obj P)) (Y :=
      obj (sheafToPresheaf J B) ((G₁ ⋙ sheafCompose J F).obj P))
      (map (sheafToPresheaf J B) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P))
⊢ IsIso (G₂.map (whiskerRight (adj₁.unit.app P) F)) ↔
    IsIso (Y := (G₁ ⋙ sheafCompose J F).obj P) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P)
have h₃ : J.W (X := (G₂ ⋙ sheafToPresheaf J B).obj (P ⋙ F)) (Y := (G₁ ⋙ sheafToPresheaf J A).obj P ⋙ F)
    (map (sheafToPresheaf J B) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P) :
      obj (sheafToPresheaf J B) (G₂.obj (P ⋙ F)) ⟶ obj (sheafToPresheaf J B) ((G₁ ⋙ sheafCompose J F).obj P)) ↔
  J.W (X := obj (sheafToPresheaf J B) ((obj (whiskeringRight Cᵒᵖ A B) F ⋙ G₂).obj P)) (Y :=
    obj (sheafToPresheaf J B) ((G₁ ⋙ sheafCompose J F).obj P))
    (map (sheafToPresheaf J B) (NatTrans.app (sheafComposeNatTrans J F adj₁ adj₂) P))
Tactic: rw [← J.W_iff_isIso_map_of_adjunction adj₂, ← J.W_sheafToPresheaf_map_iff_isIso,
    ← sheafComposeNatTrans_fac J F adj₁ adj₂,
    (W _).precomp_iff _ _ (J.W_adj_unit_app adj₂ (P ⋙ F))]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u_1
B : Type u_2
inst_1 : Category.{v_1, u_1} A
inst_2 : Category.{v_2, u_2} B
F : Functor.{v_1, v_2, u_1, u_2} A B
inst_3 : HasWeakSheafify.{v, v_1, u, u_1} J A
inst_4 : HasWeakSheafify.{v, v_2, u, u_2} J B
inst_5 : J.HasSheafCompose F
inst_6 : J.PreservesSheafification F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ A
⊢ Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J P ⋙ F) (whiskerRight (toSheafify J P) F) (Iso.inv (sheafifyComposeIso J F P)))
    (toSheafify J (P ⋙ F))
have rw : Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
  (CategoryStruct.comp (Y := sheafify J P ⋙ F)
    (CategoryStruct.comp (Y := sheafify J (P ⋙ F)) (toSheafify J (P ⋙ F)) (Iso.hom (sheafifyComposeIso J F P)))
    (Iso.inv (sheafifyComposeIso J F P)))
  (toSheafify J (P ⋙ F))
Tactic: rw [← sheafComposeIso_hom_fac, assoc, Iso.hom_inv_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u_1
B : Type u_2
inst_1 : Category.{v_1, u_1} A
inst_2 : Category.{v_2, u_2} B
F : Functor.{v_1, v_2, u_1, u_2} A B
inst_3 : HasWeakSheafify.{v, v_1, u, u_1} J A
inst_4 : HasWeakSheafify.{v, v_2, u, u_2} J B
inst_5 : J.HasSheafCompose F
inst_6 : J.PreservesSheafification F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ A
rw :
  Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J P ⋙ F)
      (CategoryStruct.comp (Y := sheafify J (P ⋙ F)) (toSheafify J (P ⋙ F)) (Iso.hom (sheafifyComposeIso J F P)))
      (Iso.inv (sheafifyComposeIso J F P)))
    (toSheafify J (P ⋙ F))
⊢ Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J P ⋙ F) (whiskerRight (toSheafify J P) F) (Iso.inv (sheafifyComposeIso J F P)))
    (toSheafify J (P ⋙ F))
have rw₁ : Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
  (CategoryStruct.comp (Y := sheafify J (P ⋙ F)) (toSheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J P ⋙ F) (Iso.hom (sheafifyComposeIso J F P))
      (Iso.inv (sheafifyComposeIso J F P))))
  (toSheafify J (P ⋙ F))
Tactic: rw [← sheafComposeIso_hom_fac, assoc, Iso.hom_inv_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u_1
B : Type u_2
inst_1 : Category.{v_1, u_1} A
inst_2 : Category.{v_2, u_2} B
F : Functor.{v_1, v_2, u_1, u_2} A B
inst_3 : HasWeakSheafify.{v, v_1, u, u_1} J A
inst_4 : HasWeakSheafify.{v, v_2, u, u_2} J B
inst_5 : J.HasSheafCompose F
inst_6 : J.PreservesSheafification F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ A
rw :
  Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J P ⋙ F)
      (CategoryStruct.comp (Y := sheafify J (P ⋙ F)) (toSheafify J (P ⋙ F)) (Iso.hom (sheafifyComposeIso J F P)))
      (Iso.inv (sheafifyComposeIso J F P)))
    (toSheafify J (P ⋙ F))
rw₁ :
  Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J (P ⋙ F)) (toSheafify J (P ⋙ F))
      (CategoryStruct.comp (Y := sheafify J P ⋙ F) (Iso.hom (sheafifyComposeIso J F P))
        (Iso.inv (sheafifyComposeIso J F P))))
    (toSheafify J (P ⋙ F))
⊢ Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J P ⋙ F) (whiskerRight (toSheafify J P) F) (Iso.inv (sheafifyComposeIso J F P)))
    (toSheafify J (P ⋙ F))
have rw₂ : toSheafify J (P ⋙ F) ≫ 𝟙 (sheafify J (P ⋙ F)) = toSheafify J (P ⋙ F)
Tactic: rw [← sheafComposeIso_hom_fac, assoc, Iso.hom_inv_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u_1
B : Type u_2
inst_1 : Category.{v_1, u_1} A
inst_2 : Category.{v_2, u_2} B
F : Functor.{v_1, v_2, u_1, u_2} A B
inst_3 : HasWeakSheafify.{v, v_1, u, u_1} J A
inst_4 : HasWeakSheafify.{v, v_2, u, u_2} J B
inst_5 : J.HasSheafCompose F
inst_6 : J.PreservesSheafification F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ A
rw :
  Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J P ⋙ F)
      (CategoryStruct.comp (Y := sheafify J (P ⋙ F)) (toSheafify J (P ⋙ F)) (Iso.hom (sheafifyComposeIso J F P)))
      (Iso.inv (sheafifyComposeIso J F P)))
    (toSheafify J (P ⋙ F))
rw₁ :
  Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J (P ⋙ F)) (toSheafify J (P ⋙ F))
      (CategoryStruct.comp (Y := sheafify J P ⋙ F) (Iso.hom (sheafifyComposeIso J F P))
        (Iso.inv (sheafifyComposeIso J F P))))
    (toSheafify J (P ⋙ F))
rw₂ : toSheafify J (P ⋙ F) ≫ 𝟙 (sheafify J (P ⋙ F)) = toSheafify J (P ⋙ F)
⊢ Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F))
    (CategoryStruct.comp (Y := sheafify J P ⋙ F) (whiskerRight (toSheafify J P) F) (Iso.inv (sheafifyComposeIso J F P)))
    (toSheafify J (P ⋙ F))
have rw₃ : Eq (α := P ⋙ F ⟶ sheafify J (P ⋙ F)) (toSheafify J (P ⋙ F)) (toSheafify J (P ⋙ F))
Tactic: rw [← sheafComposeIso_hom_fac, assoc, Iso.hom_inv_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_3
E : Type u_4
inst_1 : Category.{v_3, u_3} D
inst_2 : Category.{v_4, u_4} E
F : Functor.{v_3, v_4, u_3, u_4} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_6 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_7 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst_8 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_3, u, u_3} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst_9 : (X Y : D) → FunLike.{u_5 + 1, u_6 + 1, u_6 + 1} (FD X Y) (CD X) (CD Y)
inst_10 : (X Y : E) → FunLike.{u_7 + 1, u_8 + 1, u_8 + 1} (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory.{u_6, v_3, u_3, u_5} D FD (CC := CD)
instCCE : ConcreteCategory.{u_8, v_4, u_4, u_7} E FE (CC := CE)
inst_11 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_12 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E (FC := FE) (CC := CE))
inst_13 : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D (FC := FD) (CC := CD))
inst_14 : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E (FC := FE) (CC := CE))
inst_15 : ReflectsIsomorphisms.{v_3, u_3, u_6, u_6 + 1} (forget D (FC := FD) (CC := CD))
inst_16 : ReflectsIsomorphisms.{v_4, u_4, u_8, u_8 + 1} (forget E (FC := FE) (CC := CE))
P : Functor.{v, v_3, u, u_3} Cᵒᵖ D
this :
  Eq (α :=
    (obj (whiskeringRight Cᵒᵖ D E) F ⋙ plusPlusSheaf J E (FD := FE) (CD := CE)).obj P ⟶
      (plusPlusSheaf J D (FD := FD) (CD := CD) ⋙ sheafCompose J F).obj P)
    (NatTrans.app
      (sheafComposeNatTrans J F (plusPlusAdjunction J D (FD := FD) (CD := CD))
        (plusPlusAdjunction J E (FD := FE) (CD := CE)))
      P)
    { hom := Iso.inv (J.sheafifyCompIso F P) }
⊢ Eq (α :=
    obj (sheafToPresheaf J E) ((obj (whiskeringRight Cᵒᵖ D E) F ⋙ plusPlusSheaf J E (FD := FE) (CD := CE)).obj P) ⟶
      obj (sheafToPresheaf J E) ((plusPlusSheaf J D (FD := FD) (CD := CD) ⋙ sheafCompose J F).obj P))
    (map (sheafToPresheaf J E)
      (NatTrans.app
        (sheafComposeNatTrans J F (plusPlusAdjunction J D (FD := FD) (CD := CD))
          (plusPlusAdjunction J E (FD := FE) (CD := CE)))
        P))
    (Iso.inv (J.sheafifyCompIso F P))
have rw : Eq (α :=
  obj (sheafToPresheaf J E) ((obj (whiskeringRight Cᵒᵖ D E) F ⋙ plusPlusSheaf J E (FD := FE) (CD := CE)).obj P) ⟶
    obj (sheafToPresheaf J E) ((plusPlusSheaf J D (FD := FD) (CD := CD) ⋙ sheafCompose J F).obj P))
  (map (sheafToPresheaf J E) { hom := Iso.inv (J.sheafifyCompIso F P) }) (Iso.inv (J.sheafifyCompIso F P))
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_3
E : Type u_4
inst_1 : Category.{v_3, u_3} D
inst_2 : Category.{v_4, u_4} E
F : Functor.{v_3, v_4, u_3, u_4} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_6 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_7 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst_8 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_3, u, u_3} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst_9 : (X Y : D) → FunLike.{u_5 + 1, u_6 + 1, u_6 + 1} (FD X Y) (CD X) (CD Y)
inst_10 : (X Y : E) → FunLike.{u_7 + 1, u_8 + 1, u_8 + 1} (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory.{u_6, v_3, u_3, u_5} D FD (CC := CD)
instCCE : ConcreteCategory.{u_8, v_4, u_4, u_7} E FE (CC := CE)
inst_11 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_12 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E (FC := FE) (CC := CE))
inst_13 : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D (FC := FD) (CC := CD))
inst_14 : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E (FC := FE) (CC := CE))
inst_15 : ReflectsIsomorphisms.{v_3, u_3, u_6, u_6 + 1} (forget D (FC := FD) (CC := CD))
inst_16 : ReflectsIsomorphisms.{v_4, u_4, u_8, u_8 + 1} (forget E (FC := FE) (CC := CE))
P : Functor.{v, v_3, u, u_3} Cᵒᵖ D
⊢ Eq (α :=
    obj (sheafToPresheaf J E) ((obj (whiskeringRight Cᵒᵖ D E) F ⋙ plusPlusSheaf J E (FD := FE) (CD := CE)).obj P) ⟶
      obj (sheafToPresheaf J E) ((plusPlusSheaf J D (FD := FD) (CD := CD) ⋙ sheafCompose J F).obj P))
    (map (sheafToPresheaf J E)
      (NatTrans.app
        (sheafComposeNatTrans J F (plusPlusAdjunction J D (FD := FD) (CD := CD))
          (plusPlusAdjunction J E (FD := FE) (CD := CE)))
        P))
    (Iso.inv (J.sheafifyCompIso F P))
have suffices : Eq (α :=
  (obj (whiskeringRight Cᵒᵖ D E) F ⋙ plusPlusSheaf J E (FD := FE) (CD := CE)).obj P ⟶
    (plusPlusSheaf J D (FD := FD) (CD := CD) ⋙ sheafCompose J F).obj P)
  (NatTrans.app
    (sheafComposeNatTrans J F (plusPlusAdjunction J D (FD := FD) (CD := CD))
      (plusPlusAdjunction J E (FD := FE) (CD := CE)))
    P)
  { hom := Iso.inv (J.sheafifyCompIso F P) }
Tactic: suffices (sheafComposeNatTrans J F (plusPlusAdjunction J D) (plusPlusAdjunction J E)).app P =
    ⟨(sheafifyCompIso J F P).inv⟩ by
    rw [this]
    rfl
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u_1
B : Type u_2
inst_1 : Category.{v_1, u_1} A
inst_2 : Category.{v_2, u_2} B
F_1 : Functor.{v_1, v_2, u_1, u_2} A B
D : Type u_3
E : Type u_4
inst_3 : Category.{v_3, u_3} D
inst_4 : Category.{v_4, u_4} E
F : Functor.{v_3, v_4, u_3, u_4} D E
inst_5 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_6 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_7 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_8 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_9 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst_10 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_3, u, u_3} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst_11 : (X Y : D) → FunLike.{u_5 + 1, u_6 + 1, u_6 + 1} (FD X Y) (CD X) (CD Y)
inst_12 : (X Y : E) → FunLike.{u_7 + 1, u_8 + 1, u_8 + 1} (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory.{u_6, v_3, u_3, u_5} D FD (CC := CD)
instCCE : ConcreteCategory.{u_8, v_4, u_4, u_7} E FE (CC := CE)
inst_13 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_14 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E (FC := FE) (CC := CE))
inst_15 : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D (FC := FD) (CC := CD))
inst_16 : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E (FC := FE) (CC := CE))
inst_17 : ReflectsIsomorphisms.{v_3, u_3, u_6, u_6 + 1} (forget D (FC := FD) (CC := CD))
inst_18 : ReflectsIsomorphisms.{v_4, u_4, u_8, u_8 + 1} (forget E (FC := FE) (CC := CE))
P : Functor.{v, v_3, u, u_3} Cᵒᵖ D
⊢ IsIso (X := (obj (whiskeringRight Cᵒᵖ D E) F ⋙ plusPlusSheaf J E (FD := FE) (CD := CE)).obj P) (Y :=
    (plusPlusSheaf J D (FD := FD) (CD := CD) ⋙ sheafCompose J F).obj P)
    (NatTrans.app
      (sheafComposeNatTrans J F (plusPlusAdjunction J D (FD := FD) (CD := CD))
        (plusPlusAdjunction J E (FD := FE) (CD := CE)))
      P)
have rw : IsIso (X :=
  obj (sheafToPresheaf J E) ((obj (whiskeringRight Cᵒᵖ D E) F ⋙ plusPlusSheaf J E (FD := FE) (CD := CE)).obj P)) (Y :=
  obj (sheafToPresheaf J E) ((plusPlusSheaf J D (FD := FD) (CD := CD) ⋙ sheafCompose J F).obj P))
  (Iso.inv (J.sheafifyCompIso F P) : J.sheafify (P ⋙ F) ⟶ J.sheafify P ⋙ F)
Tactic: rw [← isIso_iff_of_reflects_iso _ (sheafToPresheaf J E),
    sheafToPresheaf_map_sheafComposeNatTrans_eq_sheafifyCompIso_inv]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.PreservesSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u_1
B : Type u_2
inst_1 : Category.{v_1, u_1} A
inst_2 : Category.{v_2, u_2} B
F_1 : Functor.{v_1, v_2, u_1, u_2} A B
D : Type u_3
E : Type u_4
inst_3 : Category.{v_3, u_3} D
inst_4 : Category.{v_4, u_4} E
F : Functor.{v_3, v_4, u_3, u_4} D E
inst_5 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_6 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_7 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_8 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_9 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst_10 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_3, u, u_3} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst_11 : (X Y : D) → FunLike.{u_5 + 1, u_6 + 1, u_6 + 1} (FD X Y) (CD X) (CD Y)
inst_12 : (X Y : E) → FunLike.{u_7 + 1, u_8 + 1, u_8 + 1} (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory.{u_6, v_3, u_3, u_5} D FD (CC := CD)
instCCE : ConcreteCategory.{u_8, v_4, u_4, u_7} E FE (CC := CE)
inst_13 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_14 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E (FC := FE) (CC := CE))
inst_15 : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D (FC := FD) (CC := CD))
inst_16 : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E (FC := FE) (CC := CE))
inst_17 : ReflectsIsomorphisms.{v_3, u_3, u_6, u_6 + 1} (forget D (FC := FD) (CC := CD))
inst_18 : ReflectsIsomorphisms.{v_4, u_4, u_8, u_8 + 1} (forget E (FC := FE) (CC := CE))
⊢ J.PreservesSheafification F
have rw : IsIso (X := obj (whiskeringRight Cᵒᵖ D E) F ⋙ plusPlusSheaf J E (FD := FE) (CD := CE)) (Y :=
  plusPlusSheaf J D (FD := FD) (CD := CD) ⋙ sheafCompose J F)
  (sheafComposeNatTrans J F (plusPlusAdjunction J D (FD := FD) (CD := CD))
    (plusPlusAdjunction J E (FD := FE) (CD := CE)))
Tactic: rw [preservesSheafification_iff_of_adjunctions_of_hasSheafCompose _ _
    (plusPlusAdjunction J D) (plusPlusAdjunction J E)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Pretopology (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasPullbacks.{v, u} C
K : Pretopology.{v, u} C
X Y : C
S : Sieve.{v, u} X
g : Y ⟶ X
R : Presieve.{v, u} X
hR : R ∈ K.coverings X
RS : R ≤ S.arrows
⊢ Membership.mem (γ := Set (Sieve.{v, u} Y)) {S : Sieve.{v, u} Y | ∃ R ∈ K.coverings Y, R ≤ S.arrows}
    (Sieve.pullback g S)
have refine : pullbackArrows g R ≤ (Sieve.pullback g S).arrows
Tactic: refine ⟨_, K.pullbacks g _ hR, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Pretopology (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasPullbacks.{v, u} C
K : Pretopology.{v, u} C
X Y : C
S : Sieve.{v, u} X
g : Y ⟶ X
R : Presieve.{v, u} X
hR : R ∈ K.coverings X
RS : R ≤ S.arrows
⊢ pullbackArrows g R ≤ (Sieve.pullback g S).arrows
have rw : Sieve.pullback g (Sieve.generate R) ≤ Sieve.pullback g S
Tactic: rw [← Sieve.generate_le_iff, Sieve.pullbackArrows_comm]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Pretopology (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasPullbacks.{v, u} C
K : Pretopology.{v, u} C
X Y : C
S : Sieve.{v, u} X
g : Y ⟶ X
R : Presieve.{v, u} X
hR : R ∈ K.coverings X
RS : R ≤ S.arrows
⊢ Sieve.pullback g (Sieve.generate R) ≤ Sieve.pullback g S
have a : Sieve.generate R ≤ S
Tactic: apply Sieve.pullback_monotone
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Pretopology (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasPullbacks.{v, u} C
K : Pretopology.{v, u} C
X : C
S : Sieve.{v, u} X
R' : Presieve.{v, u} X
hR' : R' ∈ K.coverings X
RS : R' ≤ S.arrows
R : Sieve.{v, u} X
t :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f →
      Membership.mem (γ := Set (Sieve.{v, u} Y)) {S : Sieve.{v, u} Y | ∃ R ∈ K.coverings Y, R ≤ S.arrows}
        (Sieve.pullback f R)
⊢ Membership.mem (γ := Set (Sieve.{v, u} X)) {S : Sieve.{v, u} X | ∃ R ∈ K.coverings X, R ≤ S.arrows} R
have choose : ∀ (t₁ : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → S.arrows f → Presieve.{v, u} Y)
  (t₂ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (a : S.arrows f), t₁ a ∈ K.coverings Y)
  (t₃ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (a : S.arrows f), t₁ a ≤ (Sieve.pullback f R).arrows),
  Membership.mem (γ := Set (Sieve.{v, u} X)) {S : Sieve.{v, u} X | ∃ R ∈ K.coverings X, R ≤ S.arrows} R
Tactic: choose t₁ t₂ t₃ using t
Cleared hypotheses: t
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Pretopology (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasPullbacks.{v, u} C
K : Pretopology.{v, u} C
X : C
S : Sieve.{v, u} X
R' : Presieve.{v, u} X
hR' : R' ∈ K.coverings X
RS : R' ≤ S.arrows
R : Sieve.{v, u} X
t₁ : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → S.arrows f → Presieve.{v, u} Y
t₂ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (a : S.arrows f), t₁ a ∈ K.coverings Y
t₃ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (a : S.arrows f), t₁ a ≤ (Sieve.pullback f R).arrows
⊢ Membership.mem (γ := Set (Sieve.{v, u} X)) {S : Sieve.{v, u} X | ∃ R ∈ K.coverings X, R ≤ S.arrows} R
have refine : (R'.bind fun (x : C) (f : x ⟶ X) (hf : R' f) ↦ t₁ sorry) ≤ R.arrows
Tactic: refine ⟨_, K.transitive _ _ hR' fun _ f hf => t₂ (RS _ _ hf), ?_⟩