awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
inst_2 : F.IsContinuous J K
G : Sheaf K (Type t)
⊢ Presieve.IsSheaf.{t, v₁, u₁} J (F.op ⋙ G.obj (P := Presheaf.IsSheaf K))
have rw : Presieve.IsSheaf J (F.op ⋙ G.obj (P := Presheaf.IsSheaf K) ⋙ uliftFunctor.{max u₁ v₁ u₂ v₂, t})
Tactic: rw [← Presieve.isSheaf_comp_uliftFunctor_iff.{t, max u₁ v₁ u₂ v₂}, ← isSheaf_iff_isSheaf_of_type,
    Presheaf.isSheaf_of_iso_iff (Functor.associator _ _ _), isSheaf_iff_isSheaf_of_type]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
inst_2 : F.IsContinuous J K
G : Sheaf K (Type t)
⊢ Presieve.IsSheaf J (F.op ⋙ G.obj (P := Presheaf.IsSheaf K) ⋙ uliftFunctor.{max u₁ v₁ u₂ v₂, t})
have hG : Presieve.IsSheaf K (G.obj (P := Presheaf.IsSheaf K) ⋙ uliftFunctor.{max u₁ v₁ u₂ v₂, t})
Tactic: apply isSheaf_of_isContinuous_aux.{t} J K
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
inst_2 : F.IsContinuous J K
G : Sheaf K (Type t)
⊢ Presieve.IsSheaf K (G.obj (P := Presheaf.IsSheaf K) ⋙ uliftFunctor.{max u₁ v₁ u₂ v₂, t})
have hG : Presheaf.IsSheaf K (G.obj (P := Presheaf.IsSheaf K))
Tactic: rw [Presieve.isSheaf_comp_uliftFunctor_iff, ← isSheaf_iff_isSheaf_of_type]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
A : Type u
inst_2 : Category.{t, u} A
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
H : Functor.{v₁, t, u₁, u} Cᵒᵖ A ⥤ Functor.{v₂, t, u₂, u} Dᵒᵖ A
adj : H ⊣ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) F.op
inst_3 : F.IsContinuous J K
G G' : Functor.{v₁, t, u₁, u} Cᵒᵖ A
f : G ⟶ G'
hf : J.W f
U : Functor.{v₂, t, u₂, u} Dᵒᵖ A
hU : Presheaf.IsSheaf K U
⊢ Function.Bijective (β := H.obj G ⟶ U) fun (g : H.obj G' ⟶ U) ↦ H.map f ≫ g
have rw : Function.Bijective (β := G ⟶ obj (obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) F.op) U)
  fun (g : G' ⟶ obj (obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) F.op) U) ↦ f ≫ g
Tactic: rw [adj.map_comp_bijective_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
F₁ : Functor.{v₁, v₂, u₁, u₂} C D
F₂ : Functor.{v₂, v₃, u₂, u₃} D E
F₁₂ : Functor.{v₁, v₃, u₁, u₃} C E
e : F₁ ⋙ F₂ ≅ F₁₂
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
L : GrothendieckTopology.{v₃, u₃} E
inst_3 : F₁.IsContinuous J K
inst_4 : F₂.IsContinuous K L
⊢ F₁₂.IsContinuous J L
have this : (F₁ ⋙ F₂).IsContinuous J L
Tactic: have := Functor.isContinuous_comp F₁ F₂ J K L
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
J : Precoverage.{v₁, u₁} C
K : Precoverage.{v₂, u₂} D
inst_2 : J.IsStableUnderBaseChange
inst_3 : J.HasPullbacks
inst_4 : K.IsStableUnderBaseChange
inst_5 : K.HasPullbacks
inst_6 : J.PullbacksPreservedBy F
h : J ≤ Precoverage.comap F K
x : Sheaf K.toGrothendieck (Type (max u₁ v₁ u₂ v₂))
G : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
H : Presheaf.IsSheaf K.toGrothendieck G
⊢ Presieve.IsSheaf J.toGrothendieck
    (F.op ⋙
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf K.toGrothendieck)
        (ObjectProperty.FullSubcategory.mk (P := Presheaf.IsSheaf K.toGrothendieck) G H))
have H₁ : Presieve.IsSheaf K.toGrothendieck G
Tactic: rw [isSheaf_iff_isSheaf_of_type] at H
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
J : Precoverage.{v₁, u₁} C
K : Precoverage.{v₂, u₂} D
inst_2 : J.IsStableUnderBaseChange
inst_3 : J.HasPullbacks
inst_4 : K.IsStableUnderBaseChange
inst_5 : K.HasPullbacks
inst_6 : J.PullbacksPreservedBy F
h : J ≤ Precoverage.comap F K
x : Sheaf K.toGrothendieck (Type (max u₁ v₁ u₂ v₂))
G : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
H_1 : Presheaf.IsSheaf K.toGrothendieck G
H : Presieve.IsSheaf K.toGrothendieck G
⊢ Presieve.IsSheaf J.toGrothendieck
    (F.op ⋙
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf K.toGrothendieck)
        (ObjectProperty.FullSubcategory.mk (P := Presheaf.IsSheaf K.toGrothendieck) G H_1))
have rw : ∀ (H : ∀ {X : D}, ∀ R ∈ K.toCoverage.coverings X, Presieve.IsSheafFor G R) {X : C},
  ∀ R ∈ J.toCoverage.coverings X,
    Presieve.IsSheafFor
      (F.op ⋙
        ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf K.toGrothendieck)
          (ObjectProperty.FullSubcategory.mk (P := Presheaf.IsSheaf K.toGrothendieck) G H_1))
      R
Tactic: rw [← Precoverage.toGrothendieck_toCoverage, Presieve.isSheaf_coverage] at H ⊢
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
J : Precoverage.{v₁, u₁} C
K : Precoverage.{v₂, u₂} D
inst_2 : J.IsStableUnderBaseChange
inst_3 : J.HasPullbacks
inst_4 : K.IsStableUnderBaseChange
inst_5 : K.HasPullbacks
inst_6 : J.PullbacksPreservedBy F
h : J ≤ Precoverage.comap F K
x : Sheaf K.toGrothendieck (Type (max u₁ v₁ u₂ v₂))
G : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
H_1 : Presheaf.IsSheaf K.toGrothendieck G
H : ∀ {X : D}, ∀ R ∈ K.toCoverage.coverings X, Presieve.IsSheafFor G R
X : C
R : Presieve.{v₁, u₁} X
hR : R ∈ J.toCoverage.coverings X
⊢ Presieve.IsSheafFor
    (F.op ⋙
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf K.toGrothendieck)
        (ObjectProperty.FullSubcategory.mk (P := Presheaf.IsSheaf K.toGrothendieck) G H_1))
    R
have this : F.PreservesPairwisePullbacks R
Tactic: have : F.PreservesPairwisePullbacks R := J.preservesPairwisePullbacks_of_mem hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
J : Precoverage.{v₁, u₁} C
K : Precoverage.{v₂, u₂} D
inst_2 : J.IsStableUnderBaseChange
inst_3 : J.HasPullbacks
inst_4 : K.IsStableUnderBaseChange
inst_5 : K.HasPullbacks
inst_6 : J.PullbacksPreservedBy F
h : J ≤ Precoverage.comap F K
x : Sheaf K.toGrothendieck (Type (max u₁ v₁ u₂ v₂))
G : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
H_1 : Presheaf.IsSheaf K.toGrothendieck G
H : ∀ {X : D}, ∀ R ∈ K.toCoverage.coverings X, Presieve.IsSheafFor G R
X : C
R : Presieve.{v₁, u₁} X
hR : R ∈ J.toCoverage.coverings X
this : F.PreservesPairwisePullbacks R
⊢ Presieve.IsSheafFor
    (F.op ⋙
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf K.toGrothendieck)
        (ObjectProperty.FullSubcategory.mk (P := Presheaf.IsSheaf K.toGrothendieck) G H_1))
    R
have this₁ : R.HasPairwisePullbacks
Tactic: have : R.HasPairwisePullbacks := J.hasPairwisePullbacks_of_mem hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
J : Precoverage.{v₁, u₁} C
K : Precoverage.{v₂, u₂} D
inst_2 : J.IsStableUnderBaseChange
inst_3 : J.HasPullbacks
inst_4 : K.IsStableUnderBaseChange
inst_5 : K.HasPullbacks
inst_6 : J.PullbacksPreservedBy F
h : J ≤ Precoverage.comap F K
x : Sheaf K.toGrothendieck (Type (max u₁ v₁ u₂ v₂))
G : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
H_1 : Presheaf.IsSheaf K.toGrothendieck G
H : ∀ {X : D}, ∀ R ∈ K.toCoverage.coverings X, Presieve.IsSheafFor G R
X : C
R : Presieve.{v₁, u₁} X
hR : R ∈ J.toCoverage.coverings X
this_1 : F.PreservesPairwisePullbacks R
this : R.HasPairwisePullbacks
⊢ Presieve.IsSheafFor
    (F.op ⋙
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf K.toGrothendieck)
        (ObjectProperty.FullSubcategory.mk (P := Presheaf.IsSheaf K.toGrothendieck) G H_1))
    R
have rw : Presieve.IsSheafFor
  (ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf K.toGrothendieck)
    (ObjectProperty.FullSubcategory.mk (P := Presheaf.IsSheaf K.toGrothendieck) G H_1))
  (Presieve.map F R)
Tactic: rw [Presieve.IsSheafFor.comp_iff_of_preservesPairwisePullbacks]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
A : Type u
inst_2 : Category.{t, u} A
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
inst_3 : F.PreservesOneHypercovers J K
inst_4 : J.IsGeneratedByOneHypercovers
P : Functor.{v₂, t, u₂, u} Dᵒᵖ A
hP : Presheaf.IsSheaf K P
⊢ Presheaf.IsSheaf J (F.op ⋙ P)
have rw : ∀ ⦃X : C⦄ (E : GrothendieckTopology.OneHypercover.{w, v₁, u₁} J X), Nonempty (IsLimit (E.multifork (F.op ⋙ P)))
Tactic: rw [Presheaf.isSheaf_iff_of_isGeneratedByOneHypercovers.{w}]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
inst_2 : F.PreservesOneHypercovers J K
inst_3 : J.IsGeneratedByOneHypercovers
P : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
hP : Presheaf.IsSheaf K P
⊢ Presieve.IsSheaf J
    (F.op ⋙
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf K)
        (ObjectProperty.FullSubcategory.mk (P := Presheaf.IsSheaf K) P hP))
have rw : Presheaf.IsSheaf J
  (F.op ⋙
    ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf K)
      (ObjectProperty.FullSubcategory.mk (P := Presheaf.IsSheaf K) P hP))
Tactic: rw [← isSheaf_iff_isSheaf_of_type]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
F F' : Functor.{v₁, v₂, u₁, u₂} C D
τ : F ⟶ F'
e : F ≅ F'
G : Functor.{v₂, v₃, u₂, u₃} D E
F'' : Functor.{v₁, v₁, u₁, u₁} C C
eF'' : F'' ≅ 𝟭 C
FG : Functor.{v₁, v₃, u₁, u₃} C E
eFG : F ⋙ G ≅ FG
A : Type u
inst_3 : Category.{t, u} A
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
L : GrothendieckTopology.{v₃, u₃} E
inst_4 : F.IsContinuous J K
inst_5 : F'.IsContinuous J K
⊢ Eq (α :=
    Quiver.Hom (V := Sheaf.{v₂, t, u₂, u} K A ⥤ Sheaf.{v₁, t, u₁, u} J A) (F.sheafPushforwardContinuous A J K)
      (F.sheafPushforwardContinuous A J K))
    (CategoryStruct.comp (Y := F'.sheafPushforwardContinuous A J K) (sheafPushforwardContinuousNatTrans e.inv A J K)
      (sheafPushforwardContinuousNatTrans e.hom A J K))
    (𝟙 (F.sheafPushforwardContinuous A J K))
have w : ∀ (x : Sheaf.{v₂, t, u₂, u} K A) (x_1 : Cᵒᵖ),
  Eq (α :=
    (obj (F.sheafPushforwardContinuous A J K) x).obj (P := Presheaf.IsSheaf J).obj x_1 ⟶
      (obj (F.sheafPushforwardContinuous A J K) x).obj (P := Presheaf.IsSheaf J).obj x_1)
    (NatTrans.app
      (InducedCategory.Hom.hom
        (NatTrans.app
          (CategoryStruct.comp (Y := F'.sheafPushforwardContinuous A J K)
            (sheafPushforwardContinuousNatTrans e.inv A J K) (sheafPushforwardContinuousNatTrans e.hom A J K))
          x))
      x_1)
    (NatTrans.app (InducedCategory.Hom.hom (NatTrans.app (𝟙 (F.sheafPushforwardContinuous A J K)) x)) x_1)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
F F' : Functor.{v₁, v₂, u₁, u₂} C D
τ : F ⟶ F'
e : F ≅ F'
G : Functor.{v₂, v₃, u₂, u₃} D E
F'' : Functor.{v₁, v₁, u₁, u₁} C C
eF'' : F'' ≅ 𝟭 C
FG : Functor.{v₁, v₃, u₁, u₃} C E
eFG : F ⋙ G ≅ FG
A : Type u
inst_3 : Category.{t, u} A
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
L : GrothendieckTopology.{v₃, u₃} E
inst_4 : F.IsContinuous J K
inst_5 : F'.IsContinuous J K
⊢ Eq (α :=
    Quiver.Hom (V := Sheaf.{v₂, t, u₂, u} K A ⥤ Sheaf.{v₁, t, u₁, u} J A) (F'.sheafPushforwardContinuous A J K)
      (F'.sheafPushforwardContinuous A J K))
    (CategoryStruct.comp (Y := F.sheafPushforwardContinuous A J K) (sheafPushforwardContinuousNatTrans e.hom A J K)
      (sheafPushforwardContinuousNatTrans e.inv A J K))
    (𝟙 (F'.sheafPushforwardContinuous A J K))
have w : ∀ (x : Sheaf.{v₂, t, u₂, u} K A) (x_1 : Cᵒᵖ),
  Eq (α :=
    (obj (F'.sheafPushforwardContinuous A J K) x).obj (P := Presheaf.IsSheaf J).obj x_1 ⟶
      (obj (F'.sheafPushforwardContinuous A J K) x).obj (P := Presheaf.IsSheaf J).obj x_1)
    (NatTrans.app
      (InducedCategory.Hom.hom
        (NatTrans.app
          (CategoryStruct.comp (Y := F.sheafPushforwardContinuous A J K)
            (sheafPushforwardContinuousNatTrans e.hom A J K) (sheafPushforwardContinuousNatTrans e.inv A J K))
          x))
      x_1)
    (NatTrans.app (InducedCategory.Hom.hom (NatTrans.app (𝟙 (F'.sheafPushforwardContinuous A J K)) x)) x_1)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Continuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
F : Functor.{v₁, v₂, u₁, u₂} C D
G : Functor.{v₂, v₁, u₂, u₁} D C
adj : F ⊣ G
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
inst_3 : F.IsContinuous J K
inst_4 : G.IsContinuous K J
P : Sheaf.{v₁, v₃, u₁, u₃} J E
⊢ CategoryStruct.comp (Y :=
      Functor.obj (G.sheafPushforwardContinuous E K J)
        (Functor.obj (F.sheafPushforwardContinuous E J K) (Functor.obj (G.sheafPushforwardContinuous E K J) P)))
      (NatTrans.app
          {
            app := fun (P : Sheaf.{v₂, v₃, u₂, u₃} K E) ↦
              { hom := NatTrans.app (unit (Adjunction.whiskerLeft E adj.op)) (P.obj (P := Presheaf.IsSheaf K)) },
            naturality := ⋯ }
          (Functor.obj (G.sheafPushforwardContinuous E K J) P) :
        Functor.obj (𝟭 (Sheaf.{v₂, v₃, u₂, u₃} K E)) (Functor.obj (G.sheafPushforwardContinuous E K J) P) ⟶
          Functor.obj (F.sheafPushforwardContinuous E J K ⋙ G.sheafPushforwardContinuous E K J)
            (Functor.obj (G.sheafPushforwardContinuous E K J) P))
      (Functor.map (G.sheafPushforwardContinuous E K J)
        (NatTrans.app
          {
            app := fun (P : Sheaf.{v₁, v₃, u₁, u₃} J E) ↦
              { hom := NatTrans.app (counit (Adjunction.whiskerLeft E adj.op)) (P.obj (P := Presheaf.IsSheaf J)) },
            naturality := ⋯ }
          P)) =
    𝟙 (Functor.obj (G.sheafPushforwardContinuous E K J) P)
have h : InducedCategory.Hom.hom
    (CategoryStruct.comp (Y :=
      Functor.obj (G.sheafPushforwardContinuous E K J)
        (Functor.obj (F.sheafPushforwardContinuous E J K) (Functor.obj (G.sheafPushforwardContinuous E K J) P)))
      (NatTrans.app
          {
            app := fun (P : Sheaf.{v₂, v₃, u₂, u₃} K E) ↦
              { hom := NatTrans.app (unit (Adjunction.whiskerLeft E adj.op)) (P.obj (P := Presheaf.IsSheaf K)) },
            naturality := sorry }
          (Functor.obj (G.sheafPushforwardContinuous E K J) P) :
        Functor.obj (𝟭 (Sheaf.{v₂, v₃, u₂, u₃} K E)) (Functor.obj (G.sheafPushforwardContinuous E K J) P) ⟶
          Functor.obj (F.sheafPushforwardContinuous E J K ⋙ G.sheafPushforwardContinuous E K J)
            (Functor.obj (G.sheafPushforwardContinuous E K J) P))
      (Functor.map (G.sheafPushforwardContinuous E K J)
        (NatTrans.app
          {
            app := fun (P : Sheaf.{v₁, v₃, u₁, u₃} J E) ↦
              { hom := NatTrans.app (counit (Adjunction.whiskerLeft E adj.op)) (P.obj (P := Presheaf.IsSheaf J)) },
            naturality := sorry }
          P))) =
  InducedCategory.Hom.hom (𝟙 (Functor.obj (G.sheafPushforwardContinuous E K J) P))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CoproductSheafCondition (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u_2
inst_1 : Category.{v_2, u_2} A
S : C
E : PreZeroHypercover.{w, v_1, u_1} S
inst_2 : E.HasPullbacks
c : Cofan.{w, v_1, u_1} E.X
hc : IsColimit c
huniv : IsUniversalColimit c
inst_3 : (E.sigmaOfIsColimit hc).HasPullbacks
inst_4 : ∀ (i : E.I₀), HasPullback (E.f i) ((E.sigmaOfIsColimit hc).f PUnit.unit)
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ A
inst_5 :
  PreservesLimit.{w, w, v_1, v_2, u_1, u_2} (J := Discrete E.toPreOneHypercover.I₀)
    (Discrete.functor fun (i : E.toPreOneHypercover.I₀) ↦ op (E.toPreOneHypercover.X i)) F
inst_6 :
  PreservesLimit.{w, w, v_1, v_2, u_1, u_2} (J := Discrete E.toPreOneHypercover.I₁')
    (Discrete.functor fun (i : E.toPreOneHypercover.I₁') ↦ op (E.toPreOneHypercover.Y' i)) F
⊢ IsLimit ((E.sigmaOfIsColimit hc).toPreOneHypercover.multifork F) ≃ IsLimit (E.toPreOneHypercover.multifork F)
have this : HasPullback (Cofan.IsColimit.desc hc E.f) (Cofan.IsColimit.desc hc E.f)
Tactic: have : HasPullback (Cofan.IsColimit.desc hc E.f) (Cofan.IsColimit.desc hc E.f) :=
    inferInstanceAs <| HasPullback
      ((E.sigmaOfIsColimit hc).f ⟨⟩) ((E.sigmaOfIsColimit hc).f ⟨⟩)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CoproductSheafCondition (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u_2
inst_1 : Category.{v_2, u_2} A
S : C
E : PreZeroHypercover.{w, v_1, u_1} S
inst_2 : E.HasPullbacks
c : Cofan.{w, v_1, u_1} E.X
hc : IsColimit c
huniv : IsUniversalColimit c
inst_3 : (E.sigmaOfIsColimit hc).HasPullbacks
inst_4 : ∀ (i : E.I₀), HasPullback (E.f i) ((E.sigmaOfIsColimit hc).f PUnit.unit)
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ A
inst_5 :
  PreservesLimit.{w, w, v_1, v_2, u_1, u_2} (J := Discrete E.toPreOneHypercover.I₀)
    (Discrete.functor fun (i : E.toPreOneHypercover.I₀) ↦ op (E.toPreOneHypercover.X i)) F
inst_6 :
  PreservesLimit.{w, w, v_1, v_2, u_1, u_2} (J := Discrete E.toPreOneHypercover.I₁')
    (Discrete.functor fun (i : E.toPreOneHypercover.I₁') ↦ op (E.toPreOneHypercover.Y' i)) F
this : HasPullback (Cofan.IsColimit.desc hc E.f) (Cofan.IsColimit.desc hc E.f)
c' : Cofan.{w, v_1, u_1} E.toPreOneHypercover.Y' :=
  Cofan.mk ((E.sigmaOfIsColimit hc).toPreOneHypercover.Y (i₁ := PUnit.unit) (i₂ := PUnit.unit) PUnit.unit)
    fun (b : E.toPreOneHypercover.I₁') ↦
    pullback.map
      (E.f b.fst (β := fun (i : E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀) ↦ E.toPreOneHypercover.I₁ i.1 i.2).1)
      (E.f b.fst (β := fun (i : E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀) ↦ E.toPreOneHypercover.I₁ i.1 i.2).2)
      ((E.sigmaOfIsColimit hc).f PUnit.unit) ((E.sigmaOfIsColimit hc).f PUnit.unit)
      (c.inj
        b.fst (β := fun (i : E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀) ↦ E.toPreOneHypercover.I₁ i.1 i.2).1)
      (c.inj
        b.fst (β := fun (i : E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀) ↦ E.toPreOneHypercover.I₁ i.1 i.2).2)
      (𝟙 S) ⋯ ⋯
equiv : E.toPreOneHypercover.I₁' ≃ E.I₀ × E.I₀ := Equiv.sigmaPUnit (E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀)
⊢ IsColimit c'
have refine : Nonempty
  (IsColimit.{w, v_1, w, u_1} (J := Discrete (E.I₀ × E.I₀)) (F :=
    Discrete.functor fun (i : E.I₀ × E.I₀) ↦
      E.toPreOneHypercover.Y' ((equiv.symm : (a : E.I₀ × E.I₀) → E.toPreOneHypercover.I₁') i))
    (Cofan.mk c'.pt fun (i : E.I₀ × E.I₀) ↦ c'.inj ((equiv.symm : (a : E.I₀ × E.I₀) → E.toPreOneHypercover.I₁') i)))
Tactic: refine (c'.isColimitEquivOfEquiv equiv.symm).symm (Nonempty.some ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CoproductSheafCondition (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u_2
inst_1 : Category.{v_2, u_2} A
S : C
E : PreZeroHypercover.{w, v_1, u_1} S
inst_2 : E.HasPullbacks
c : Cofan.{w, v_1, u_1} E.X
hc : IsColimit c
huniv : IsUniversalColimit c
inst_3 : (E.sigmaOfIsColimit hc).HasPullbacks
inst_4 : ∀ (i : E.I₀), HasPullback (E.f i) ((E.sigmaOfIsColimit hc).f PUnit.unit)
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ A
inst_5 :
  PreservesLimit.{w, w, v_1, v_2, u_1, u_2} (J := Discrete E.toPreOneHypercover.I₀)
    (Discrete.functor fun (i : E.toPreOneHypercover.I₀) ↦ op (E.toPreOneHypercover.X i)) F
inst_6 :
  PreservesLimit.{w, w, v_1, v_2, u_1, u_2} (J := Discrete E.toPreOneHypercover.I₁')
    (Discrete.functor fun (i : E.toPreOneHypercover.I₁') ↦ op (E.toPreOneHypercover.Y' i)) F
this : HasPullback (Cofan.IsColimit.desc hc E.f) (Cofan.IsColimit.desc hc E.f)
c' : Cofan.{w, v_1, u_1} E.toPreOneHypercover.Y' :=
  Cofan.mk ((E.sigmaOfIsColimit hc).toPreOneHypercover.Y (i₁ := PUnit.unit) (i₂ := PUnit.unit) PUnit.unit)
    fun (b : E.toPreOneHypercover.I₁') ↦
    pullback.map
      (E.f b.fst (β := fun (i : E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀) ↦ E.toPreOneHypercover.I₁ i.1 i.2).1)
      (E.f b.fst (β := fun (i : E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀) ↦ E.toPreOneHypercover.I₁ i.1 i.2).2)
      ((E.sigmaOfIsColimit hc).f PUnit.unit) ((E.sigmaOfIsColimit hc).f PUnit.unit)
      (c.inj
        b.fst (β := fun (i : E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀) ↦ E.toPreOneHypercover.I₁ i.1 i.2).1)
      (c.inj
        b.fst (β := fun (i : E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀) ↦ E.toPreOneHypercover.I₁ i.1 i.2).2)
      (𝟙 S) ⋯ ⋯
equiv : E.toPreOneHypercover.I₁' ≃ E.I₀ × E.I₀ := Equiv.sigmaPUnit (E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀)
⊢ IsLimit ((E.sigmaOfIsColimit hc).toPreOneHypercover.multifork F) ≃ IsLimit (E.toPreOneHypercover.multifork F)
have have : (hc' : IsColimit c') →
  IsLimit ((E.sigmaOfIsColimit hc).toPreOneHypercover.multifork F) ≃ IsLimit (E.toPreOneHypercover.multifork F)
Tactic: have hc' : IsColimit c' := by
    refine (c'.isColimitEquivOfEquiv equiv.symm).symm (Nonempty.some ?_)
    exact IsUniversalColimit.nonempty_isColimit_prod_of_isPullback
      huniv huniv E.f E.f ((E.sigmaOfIsColimit hc).f ⟨⟩) ((E.sigmaOfIsColimit hc).f ⟨⟩)
      (fun i j ↦ .of_hasPullback _ _) (.of_hasPullback _ _) (.refl _) (by simp) (by simp)
      (by simp [c', equiv, Equiv.sigmaPUnit]) (by simp [c', equiv, Equiv.sigmaPUnit])