awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
hI : IsInitial.{v, u} I
α : Type u_1
inst_1 : Small.{w, u_1} α
X : α → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
inst_2 : (ofArrows X c.inj).HasPairwisePullbacks
inst_3 : HasInitial.{v, u} C
inst_4 : ∀ (i : α), Mono (c.inj i)
hd :
  Pairwise fun (i j : α) ↦ IsPullback (P := initial.{v, u} C) (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
hF' : IsSheafFor F (ofArrows X c.inj)
this : HasCoproduct.{u_1, v, u} X
⊢ ∀ (y : Equalizer.Presieve.Arrows.FirstObj F X),
    Eq (α := (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.firstMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X)) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
            y :
          (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.secondMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X)) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
            y :
          (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj)) →
      ∃! x : F.obj (op c.pt),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.forkMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (F.obj (op c.pt))) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X))
            x =
          y

Tactic:
rw [Equalizer.Presieve.Arrows.sheaf_condition, Limits.Types.type_equalizer_iff_unique] at hF'

Premises used: CategoryTheory.Equalizer.Presieve.Arrows.sheaf_condition, CategoryTheory.Limits.Types.type_equalizer_iff_unique
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
hI : IsInitial.{v, u} I
α : Type u_1
inst_1 : Small.{w, u_1} α
X : α → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
inst_2 : (ofArrows X c.inj).HasPairwisePullbacks
inst_3 : HasInitial.{v, u} C
inst_4 : ∀ (i : α), Mono (c.inj i)
hd :
  Pairwise fun (i j : α) ↦ IsPullback (P := initial.{v, u} C) (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
hF' : IsSheafFor F (ofArrows X c.inj)
this : HasCoproduct.{u_1, v, u} X
hF'₁ :
  ∀ (y : Equalizer.Presieve.Arrows.FirstObj F X),
    Eq (α := (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.firstMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X)) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
            y :
          (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.secondMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X)) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
            y :
          (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj)) →
      ∃! x : F.obj (op c.pt),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.forkMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (F.obj (op c.pt))) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X))
            x =
          y
⊢ HasCoproduct.{u_1, v, u} X

Tactic:
rw [Equalizer.Presieve.Arrows.sheaf_condition, Limits.Types.type_equalizer_iff_unique] at hF'

Premises used: CategoryTheory.Equalizer.Presieve.Arrows.sheaf_condition, CategoryTheory.Limits.Types.type_equalizer_iff_unique
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
hI : IsInitial.{v, u} I
α : Type u_1
inst_1 : Small.{w, u_1} α
X : α → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
inst_2 : (ofArrows X c.inj).HasPairwisePullbacks
inst_3 : HasInitial.{v, u} C
inst_4 : ∀ (i : α), Mono (c.inj i)
hd :
  Pairwise fun (i j : α) ↦ IsPullback (P := initial.{v, u} C) (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
hF' :
  ∀ (y : Equalizer.Presieve.Arrows.FirstObj F X),
    Eq (α := (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.firstMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X)) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
            y :
          (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.secondMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X)) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
            y :
          (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj)) →
      ∃! x : F.obj (op c.pt),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.forkMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (F.obj (op c.pt))) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X))
            x =
          y
this : HasCoproduct.{u_1, v, u} X
⊢ ∀ (b : (fun (X : Type w) ↦ X) (∏ᶜ fun (b : α) ↦ F.obj ((fun (x : α) ↦ op (X x)) b))),
    ∃! a : (fun (X : Type w) ↦ X) (F.obj (op c.pt)),
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.forkMap F X c.inj) :
              TypeCat.Fun (F.obj (op c.pt)) (∏ᶜ fun (b : α) ↦ F.obj ((fun (x : α) ↦ op (X x)) b))) :
            (a : (fun (X : Type w) ↦ X) (F.obj (op c.pt))) →
              (fun (X : Type w) ↦ X) (∏ᶜ fun (b : α) ↦ F.obj ((fun (x : α) ↦ op (X x)) b)))
          a =
        b

Tactic:
fun b ↦ hF' b (ConcreteCategory.congr_hom (firstMap_eq_secondMap F hF hI c hd) b)
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
hI : IsInitial.{v, u} I
α : Type u_1
inst_1 : Small.{w, u_1} α
X : α → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
inst_2 : (ofArrows X c.inj).HasPairwisePullbacks
inst_3 : HasInitial.{v, u} C
inst_4 : ∀ (i : α), Mono (c.inj i)
hd :
  Pairwise fun (i j : α) ↦ IsPullback (P := initial.{v, u} C) (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
hF' :
  ∀ (y : Equalizer.Presieve.Arrows.FirstObj F X),
    Eq (α := (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.firstMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X)) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
            y :
          (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.secondMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X)) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj))
            y :
          (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.SecondObj F X c.inj)) →
      ∃! x : F.obj (op c.pt),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.forkMap F X c.inj) :
              (a : (fun (X : Type w) ↦ X) (F.obj (op c.pt))) →
                (fun (X : Type w) ↦ X) (Equalizer.Presieve.Arrows.FirstObj F X))
            x =
          y
this : HasCoproduct.{u_1, v, u} X
⊢ ∀ (b : (fun (X : Type w) ↦ X) (∏ᶜ fun (b : α) ↦ F.obj ((fun (x : α) ↦ op (X x)) b))),
    ∃! a : (fun (X : Type w) ↦ X) (F.obj (op c.pt)),
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (Equalizer.Presieve.Arrows.forkMap F X c.inj) :
              TypeCat.Fun (F.obj (op c.pt)) (∏ᶜ fun (b : α) ↦ F.obj ((fun (x : α) ↦ op (X x)) b))) :
            (a : (fun (X : Type w) ↦ X) (F.obj (op c.pt))) →
              (fun (X : Type w) ↦ X) (∏ᶜ fun (b : α) ↦ F.obj ((fun (x : α) ↦ op (X x)) b)))
          a =
        b

Tactic:
exact fun b ↦ hF' b (ConcreteCategory.congr_hom (firstMap_eq_secondMap F hF hI c hd) b)
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
hI : IsInitial.{v, u} I
α : Type u_1
inst_1 : Small.{w, u_1} α
X : α → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
inst_2 : (ofArrows X c.inj).HasPairwisePullbacks
inst_3 : HasInitial.{v, u} C
inst_4 : ∀ (i : α), Mono (c.inj i)
hd :
  Pairwise fun (i j : α) ↦ IsPullback (P := initial.{v, u} C) (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
⊢ IsSheafFor F (ofArrows X c.inj) ↔
    PreservesLimit.{u_1, u_1, v, w, u, w + 1} (J := Discrete α) (Discrete.functor fun (x : α) ↦ op (X x)) F

Tactic:
⟨fun hF' ↦ preservesProduct_of_isSheafFor _ hF hI c hc hd hF',
    fun _ ↦ isSheafFor_of_preservesProduct F c hc⟩

Premises used: CategoryTheory.Presieve.preservesProduct_of_isSheafFor, CategoryTheory.Presieve.isSheafFor_of_preservesProduct, Iff.intro
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
inst_3 : ∀ (X : Sheaf J (Type v)), PreservesLimit F.op (X.obj (P := Presheaf.IsSheaf J))
c : Cocone F
hc : IsColimit c
suffices : IsLimit (F := (F ⋙ J.yoneda).op ⋙ coyoneda) (mapCone coyoneda (J.yoneda.mapCocone c).op)
⊢ IsColimit (J.yoneda.mapCocone c)

Tactic:
suffices IsLimit (coyoneda.mapCone (J.yoneda.mapCocone c).op) from
      isColimitOfOp (isLimitOfReflects _ this)
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
inst_3 : ∀ (X : Sheaf J (Type v)), PreservesLimit F.op (X.obj (P := Presheaf.IsSheaf J))
c : Cocone F
hc : IsColimit c
t :
  (k : Sheaf J (Type v)) →
    IsLimit (F := ((F ⋙ J.yoneda).op ⋙ coyoneda) ⋙ obj (evaluation (Sheaf J (Type v)) (Type (max u v))) k)
      (mapCone (obj (evaluation (Sheaf J (Type v)) (Type (max u v))) k) (mapCone coyoneda (J.yoneda.mapCocone c).op))
⊢ IsLimit (F := (F ⋙ J.yoneda).op ⋙ coyoneda) (mapCone coyoneda (J.yoneda.mapCocone c).op)

Tactic:
apply evaluationJointlyReflectsLimits

Premises used: CategoryTheory.Limits.evaluationJointlyReflectsLimits
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
inst_3 : ∀ (X : Sheaf J (Type v)), PreservesLimit F.op (X.obj (P := Presheaf.IsSheaf J))
c : Cocone F
hc : IsColimit c
X : Sheaf J (Type v)
t : IsLimit ((X.obj (P := Presheaf.IsSheaf J) ⋙ uliftFunctor.{u, v}).mapCone c.op)
⊢ IsLimit (F := ((F ⋙ J.yoneda).op ⋙ coyoneda) ⋙ obj (evaluation (Sheaf J (Type v)) (Type (max u v))) X)
    (mapCone (obj (evaluation (Sheaf J (Type v)) (Type (max u v))) X) (mapCone coyoneda (J.yoneda.mapCocone c).op))

Tactic:
suffices IsLimit ((X.obj ⋙ uliftFunctor).mapCone c.op) from IsLimit.mapConeEquiv
      (isoWhiskerRight (J.yonedaOpCompCoyoneda) ((evaluation _ _ ).obj X)).symm this
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
inst_3 : ∀ (X : Sheaf J (Type v)), PreservesLimit F.op (X.obj (P := Presheaf.IsSheaf J))
c : Cocone F
hc : IsColimit c
X : Sheaf J (Type v)
⊢ IsLimit ((X.obj (P := Presheaf.IsSheaf J) ⋙ uliftFunctor.{u, v}).mapCone c.op)

Tactic:
exact isLimitOfPreserves _ hc.op

Premises used: CategoryTheory.Limits.IsColimit.op, CategoryTheory.Limits.isLimitOfPreserves
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
inst_3 : ∀ (X : Sheaf J (Type (max v v'))), PreservesLimit F.op (X.obj (P := Presheaf.IsSheaf J))
c : Cocone F
hc : IsColimit c
suffices :
  IsLimit (F := (F ⋙ GrothendieckTopology.uliftYoneda.{v', v, u} J).op ⋙ coyoneda)
    (mapCone coyoneda ((GrothendieckTopology.uliftYoneda.{v', v, u} J).mapCocone c).op)
⊢ IsColimit ((GrothendieckTopology.uliftYoneda.{v', v, u} J).mapCocone c)

Tactic:
suffices IsLimit (coyoneda.mapCone (J.uliftYoneda.mapCocone c).op) from
      isColimitOfOp (isLimitOfReflects _ this)
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
inst_3 : ∀ (X : Sheaf J (Type (max v v'))), PreservesLimit F.op (X.obj (P := Presheaf.IsSheaf J))
c : Cocone F
hc : IsColimit c
t :
  (k : Sheaf J (Type (max v v'))) →
    IsLimit (F :=
      ((F ⋙ GrothendieckTopology.uliftYoneda.{v', v, u} J).op ⋙ coyoneda) ⋙
        obj (evaluation (Sheaf J (Type (max v v'))) (Type (max (max u v) v'))) k)
      (mapCone (obj (evaluation (Sheaf J (Type (max v v'))) (Type (max (max u v) v'))) k)
        (mapCone coyoneda ((GrothendieckTopology.uliftYoneda.{v', v, u} J).mapCocone c).op))
⊢ IsLimit (F := (F ⋙ GrothendieckTopology.uliftYoneda.{v', v, u} J).op ⋙ coyoneda)
    (mapCone coyoneda ((GrothendieckTopology.uliftYoneda.{v', v, u} J).mapCocone c).op)

Tactic:
apply evaluationJointlyReflectsLimits

Premises used: CategoryTheory.Limits.evaluationJointlyReflectsLimits
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
inst_3 : ∀ (X : Sheaf J (Type (max v v'))), PreservesLimit F.op (X.obj (P := Presheaf.IsSheaf J))
c : Cocone F
hc : IsColimit c
X : Sheaf J (Type (max v v'))
t : IsLimit ((X.obj (P := Presheaf.IsSheaf J) ⋙ uliftFunctor.{u, max v v'}).mapCone c.op)
⊢ IsLimit (F :=
    ((F ⋙ GrothendieckTopology.uliftYoneda.{v', v, u} J).op ⋙ coyoneda) ⋙
      obj (evaluation (Sheaf J (Type (max v v'))) (Type (max (max u v) v'))) X)
    (mapCone (obj (evaluation (Sheaf J (Type (max v v'))) (Type (max (max u v) v'))) X)
      (mapCone coyoneda ((GrothendieckTopology.uliftYoneda.{v', v, u} J).mapCocone c).op))

Tactic:
suffices IsLimit ((X.obj ⋙ uliftFunctor).mapCone c.op) from IsLimit.mapConeEquiv
      (isoWhiskerRight (J.uliftYonedaOpCompCoyoneda) ((evaluation _ _ ).obj X)).symm this
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
inst_3 : ∀ (X : Sheaf J (Type (max v v'))), PreservesLimit F.op (X.obj (P := Presheaf.IsSheaf J))
c : Cocone F
hc : IsColimit c
X : Sheaf J (Type (max v v'))
⊢ IsLimit ((X.obj (P := Presheaf.IsSheaf J) ⋙ uliftFunctor.{u, max v v'}).mapCone c.op)

Tactic:
exact isLimitOfPreserves _ hc.op

Premises used: CategoryTheory.Limits.IsColimit.op, CategoryTheory.Limits.isLimitOfPreserves
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
⊢ PreservesLimitsOfShape K (J.yoneda ⋙ sheafToPresheaf J (Type v))

Tactic:
have : PreservesLimitsOfShape K (J.yoneda ⋙ sheafToPresheaf J (Type v)) :=
      preservesLimitsOfShape_of_natIso (yonedaCompSheafToPresheaf J).symm
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
this : PreservesLimitsOfShape K (J.yoneda ⋙ sheafToPresheaf J (Type v))
⊢ PreservesLimit F J.yoneda

Tactic:
preservesLimit_of_reflects_of_preserves (F := J.yoneda) (G := sheafToPresheaf ..)

Premises used: CategoryTheory.GrothendieckTopology.yoneda, CategoryTheory.Limits.preservesLimit_of_reflects_of_preserves, CategoryTheory.sheafToPresheaf
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
this : PreservesLimitsOfShape K (J.yoneda ⋙ sheafToPresheaf J (Type v))
⊢ PreservesLimit F J.yoneda

Tactic:
exact preservesLimit_of_reflects_of_preserves (F := J.yoneda) (G := sheafToPresheaf ..)

Premises used: CategoryTheory.GrothendieckTopology.yoneda, CategoryTheory.Limits.preservesLimit_of_reflects_of_preserves, CategoryTheory.sheafToPresheaf
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
⊢ PreservesLimitsOfShape K (GrothendieckTopology.uliftYoneda.{v', v, u} J ⋙ sheafToPresheaf J (Type (max v v')))

Tactic:
have : PreservesLimitsOfShape K (uliftYoneda J ⋙ sheafToPresheaf J (Type max v v')) :=
      preservesLimitsOfShape_of_natIso (uliftYonedaCompSheafToPresheaf J).symm
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
this : PreservesLimitsOfShape K (GrothendieckTopology.uliftYoneda.{v', v, u} J ⋙ sheafToPresheaf J (Type (max v v')))
⊢ PreservesLimit F (GrothendieckTopology.uliftYoneda.{v', v, u} J)

Tactic:
preservesLimit_of_reflects_of_preserves (uliftYoneda J)
      (sheafToPresheaf ..)

Premises used: CategoryTheory.GrothendieckTopology.uliftYoneda, CategoryTheory.Limits.preservesLimit_of_reflects_of_preserves, CategoryTheory.sheafToPresheaf
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
this : PreservesLimitsOfShape K (GrothendieckTopology.uliftYoneda.{v', v, u} J ⋙ sheafToPresheaf J (Type (max v v')))
⊢ PreservesLimit F (GrothendieckTopology.uliftYoneda.{v', v, u} J)

Tactic:
exact preservesLimit_of_reflects_of_preserves (uliftYoneda J)
      (sheafToPresheaf ..)

Premises used: CategoryTheory.GrothendieckTopology.uliftYoneda, CategoryTheory.Limits.preservesLimit_of_reflects_of_preserves, CategoryTheory.sheafToPresheaf
-- Module: Mathlib.CategoryTheory.Sites.PreservesLimits
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
K : Type u_1
inst_2 : Category.{v_1, u_1} K
F : CategoryTheory.Functor.{v_1, v, u_1, u} K C
inst_3 : ∀ (X : Sheaf J (Type v)), PreservesLimit F.op (X.obj (P := Presheaf.IsSheaf J))
⊢ PreservesColimit F J.yoneda

Tactic:
preservesColimit_of_natIso _ J.uliftYonedaIsoYoneda

Premises used: CategoryTheory.GrothendieckTopology.uliftYonedaIsoYoneda, CategoryTheory.Limits.preservesColimit_of_natIso
-- Module: Mathlib.CategoryTheory.Sites.PreservesLocallyBijective
Goal:
C : Type u_1
D : Type u_2
A : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Category.{v_3, u_3} A
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
H : Functor.{v_1, v_2, u_1, u_2} C D
F G : Functor.{v_2, v_3, u_2, u_3} Dᵒᵖ A
f : F ⟶ G
FA : A → A → Type u_4
CA : A → Type u_5
inst_3 : (X Y : A) → FunLike.{u_4 + 1, u_5 + 1, u_5 + 1} (FA X Y) (CA X) (CA Y)
inst_4 : ConcreteCategory.{u_5, v_3, u_3, u_4} A FA (CC := CA)
hH : CoverPreserving J K H
inst_5 : H.IsCoverDense K
inst_6 : IsLocallyInjective (FD := FA) (CD := CA) J (H.op.whiskerLeft f)
X : Dᵒᵖ
a b : ToType.{u_5, v_3, u_3, u_4} (FC := FA) (CC := CA) (F.obj X)
h :
  Eq (α := CA (G.obj X))
    ((ConcreteCategory.hom (FC := FA) (CC := CA) (f.app X) : (a : CA (F.obj X)) → CA (G.obj X)) a : CA (G.obj X))
    ((ConcreteCategory.hom (FC := FA) (CC := CA) (f.app X) : (a : CA (F.obj X)) → CA (G.obj X)) b : CA (G.obj X))
h₁ :
  ∀ ⦃Y : D⦄ ⦃f : Y ⟶ Opposite.unop X⦄,
    Sieve.arrows (Sieve.coverByImage H (Opposite.unop X)) f →
      Sieve.pullback f (equalizerSieve (FD := FA) (CD := CA) a b) ∈ (K : (a : D) → Set (Sieve a)) Y
⊢ Membership.mem (γ := Set (Sieve.{v_2, u_2} (Opposite.unop X))) ((K : (a : D) → Set (Sieve a)) (Opposite.unop X))
    (equalizerSieve (FD := FA) (CD := CA) a b)

Tactic:
apply K.transitive (H.is_cover_of_isCoverDense K X.unop)

Premises used: CategoryTheory.GrothendieckTopology.transitive, CategoryTheory.Functor.is_cover_of_isCoverDense, Opposite.unop
-- Module: Mathlib.CategoryTheory.Sites.PreservesLocallyBijective
Goal:
C : Type u_1
D : Type u_2
A : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Category.{v_3, u_3} A
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
H : Functor.{v_1, v_2, u_1, u_2} C D
F G : Functor.{v_2, v_3, u_2, u_3} Dᵒᵖ A
f : F ⟶ G
FA : A → A → Type u_4
CA : A → Type u_5
inst_3 : (X Y : A) → FunLike.{u_4 + 1, u_5 + 1, u_5 + 1} (FA X Y) (CA X) (CA Y)
inst_4 : ConcreteCategory.{u_5, v_3, u_3, u_4} A FA (CC := CA)
hH : CoverPreserving J K H
inst_5 : H.IsCoverDense K
inst_6 : IsLocallyInjective (FD := FA) (CD := CA) J (H.op.whiskerLeft f)
X : Dᵒᵖ
a b : ToType.{u_5, v_3, u_3, u_4} (FC := FA) (CC := CA) (F.obj X)
h :
  Eq (α := CA (G.obj X))
    ((ConcreteCategory.hom (FC := FA) (CC := CA) (f.app X) : (a : CA (F.obj X)) → CA (G.obj X)) a : CA (G.obj X))
    ((ConcreteCategory.hom (FC := FA) (CC := CA) (f.app X) : (a : CA (F.obj X)) → CA (G.obj X)) b : CA (G.obj X))
Y : D
g : Y ⟶ Opposite.unop X
h₁ :
  ∀ (Z : C) (lift : Y ⟶ H.obj Z) (m : H.obj Z ⟶ Opposite.unop X) (fac : lift ≫ m = g),
    Sieve.pullback g (equalizerSieve (FD := FA) (CD := CA) a b) ∈ (K : (a : D) → Set (Sieve a)) Y
⊢ Sieve.arrows (Sieve.coverByImage H (Opposite.unop X)) g →
    Sieve.pullback g (equalizerSieve (FD := FA) (CD := CA) a b) ∈ (K : (a : D) → Set (Sieve a)) Y

Tactic:
⟨⟨Z, lift, m, fac⟩⟩