l3lab/ntp-mathlib huggingface dataset:
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
  haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
  

Proof state:
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁴ : PreservesLimits (forget D)
inst✝³ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝² : SmallCategory K
inst✝¹ : FinCategory K
inst✝ : HasLimitsOfShape K D
e : K ≌ AsSmall (FinCategory.AsType K) := (FinCategory.equivAsType K).symm.trans AsSmall.equiv
this✝ : HasLimitsOfShape (AsSmall (FinCategory.AsType K)) D
this : FinCategory (AsSmall (FinCategory.AsType K))
⊢ PreservesLimitsOfShape K (presheafToSheaf J D)

Next tactic:
refine @preservesLimitsOfShapeOfEquiv _ _ _ _ _ _ _ _ e.symm _ (show _ from ?_)
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
  haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
  refine @preservesLimitsOfShapeOfEquiv _ _ _ _ _ _ _ _ e.symm _ (show _ from ?_)
  

Proof state:
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁴ : PreservesLimits (forget D)
inst✝³ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝² : SmallCategory K
inst✝¹ : FinCategory K
inst✝ : HasLimitsOfShape K D
e : K ≌ AsSmall (FinCategory.AsType K) := (FinCategory.equivAsType K).symm.trans AsSmall.equiv
this✝ : HasLimitsOfShape (AsSmall (FinCategory.AsType K)) D
this : FinCategory (AsSmall (FinCategory.AsType K))
⊢ PreservesLimitsOfShape (AsSmall (FinCategory.AsType K)) (presheafToSheaf J D)

Next tactic:
constructor
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
  haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
  refine @preservesLimitsOfShapeOfEquiv _ _ _ _ _ _ _ _ e.symm _ (show _ from ?_)
  constructor; 

Proof state:
case preservesLimit
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁴ : PreservesLimits (forget D)
inst✝³ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝² : SmallCategory K
inst✝¹ : FinCategory K
inst✝ : HasLimitsOfShape K D
e : K ≌ AsSmall (FinCategory.AsType K) := (FinCategory.equivAsType K).symm.trans AsSmall.equiv
this✝ : HasLimitsOfShape (AsSmall (FinCategory.AsType K)) D
this : FinCategory (AsSmall (FinCategory.AsType K))
⊢ autoParam ({K_1 : AsSmall (FinCategory.AsType K) ⥤ Cᵒᵖ ⥤ D} → PreservesLimit K_1 (presheafToSheaf J D)) _auto✝

Next tactic:
intro F
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
  haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
  refine @preservesLimitsOfShapeOfEquiv _ _ _ _ _ _ _ _ e.symm _ (show _ from ?_)
  constructor; intro F; 

Proof state:
case preservesLimit
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁴ : PreservesLimits (forget D)
inst✝³ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝² : SmallCategory K
inst✝¹ : FinCategory K
inst✝ : HasLimitsOfShape K D
e : K ≌ AsSmall (FinCategory.AsType K) := (FinCategory.equivAsType K).symm.trans AsSmall.equiv
this✝ : HasLimitsOfShape (AsSmall (FinCategory.AsType K)) D
this : FinCategory (AsSmall (FinCategory.AsType K))
F : AsSmall (FinCategory.AsType K) ⥤ Cᵒᵖ ⥤ D
⊢ PreservesLimit F (presheafToSheaf J D)

Next tactic:
constructor
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
  haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
  refine @preservesLimitsOfShapeOfEquiv _ _ _ _ _ _ _ _ e.symm _ (show _ from ?_)
  constructor; intro F; constructor; 

Proof state:
case preservesLimit.preserves
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁴ : PreservesLimits (forget D)
inst✝³ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝² : SmallCategory K
inst✝¹ : FinCategory K
inst✝ : HasLimitsOfShape K D
e : K ≌ AsSmall (FinCategory.AsType K) := (FinCategory.equivAsType K).symm.trans AsSmall.equiv
this✝ : HasLimitsOfShape (AsSmall (FinCategory.AsType K)) D
this : FinCategory (AsSmall (FinCategory.AsType K))
F : AsSmall (FinCategory.AsType K) ⥤ Cᵒᵖ ⥤ D
⊢ {c : Cone F} → IsLimit c → IsLimit ((presheafToSheaf J D).mapCone c)

Next tactic:
intro S hS
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
  haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
  refine @preservesLimitsOfShapeOfEquiv _ _ _ _ _ _ _ _ e.symm _ (show _ from ?_)
  constructor; intro F; constructor; intro S hS
  

Proof state:
case preservesLimit.preserves
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁴ : PreservesLimits (forget D)
inst✝³ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝² : SmallCategory K
inst✝¹ : FinCategory K
inst✝ : HasLimitsOfShape K D
e : K ≌ AsSmall (FinCategory.AsType K) := (FinCategory.equivAsType K).symm.trans AsSmall.equiv
this✝ : HasLimitsOfShape (AsSmall (FinCategory.AsType K)) D
this : FinCategory (AsSmall (FinCategory.AsType K))
F : AsSmall (FinCategory.AsType K) ⥤ Cᵒᵖ ⥤ D
S : Cone F
hS : IsLimit S
⊢ IsLimit ((presheafToSheaf J D).mapCone S)

Next tactic:
apply isLimitOfReflects (sheafToPresheaf J D)
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
  haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
  refine @preservesLimitsOfShapeOfEquiv _ _ _ _ _ _ _ _ e.symm _ (show _ from ?_)
  constructor; intro F; constructor; intro S hS
  apply isLimitOfReflects (sheafToPresheaf J D)
  

Proof state:
case preservesLimit.preserves.t
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁴ : PreservesLimits (forget D)
inst✝³ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝² : SmallCategory K
inst✝¹ : FinCategory K
inst✝ : HasLimitsOfShape K D
e : K ≌ AsSmall (FinCategory.AsType K) := (FinCategory.equivAsType K).symm.trans AsSmall.equiv
this✝ : HasLimitsOfShape (AsSmall (FinCategory.AsType K)) D
this : FinCategory (AsSmall (FinCategory.AsType K))
F : AsSmall (FinCategory.AsType K) ⥤ Cᵒᵖ ⥤ D
S : Cone F
hS : IsLimit S
⊢ IsLimit ((sheafToPresheaf J D).mapCone ((presheafToSheaf J D).mapCone S))

Next tactic:
have : ReflectsLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) (forget D) :=
    reflectsLimitsOfShapeOfReflectsIsomorphisms
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
  haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
  refine @preservesLimitsOfShapeOfEquiv _ _ _ _ _ _ _ _ e.symm _ (show _ from ?_)
  constructor; intro F; constructor; intro S hS
  apply isLimitOfReflects (sheafToPresheaf J D)
  have : ReflectsLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) (forget D) :=
    reflectsLimitsOfShapeOfReflectsIsomorphisms
  -- porting note: the mathlib proof was by `apply is_limit_of_preserves (J.sheafification D) hS`
  

Proof state:
case preservesLimit.preserves.t
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁴ : PreservesLimits (forget D)
inst✝³ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝² : SmallCategory K
inst✝¹ : FinCategory K
inst✝ : HasLimitsOfShape K D
e : K ≌ AsSmall (FinCategory.AsType K) := (FinCategory.equivAsType K).symm.trans AsSmall.equiv
this✝¹ : HasLimitsOfShape (AsSmall (FinCategory.AsType K)) D
this✝ : FinCategory (AsSmall (FinCategory.AsType K))
F : AsSmall (FinCategory.AsType K) ⥤ Cᵒᵖ ⥤ D
S : Cone F
hS : IsLimit S
this : ReflectsLimitsOfShape (AsSmall (FinCategory.AsType K)) (forget D)
⊢ IsLimit ((sheafToPresheaf J D).mapCone ((presheafToSheaf J D).mapCone S))

Next tactic:
have : PreservesLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K))
      (presheafToSheaf J D ⋙ sheafToPresheaf J D) :=
    preservesLimitsOfShapeOfNatIso (J.sheafificationIsoPresheafToSheafCompSheafToPreasheaf D)
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
  haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
  refine @preservesLimitsOfShapeOfEquiv _ _ _ _ _ _ _ _ e.symm _ (show _ from ?_)
  constructor; intro F; constructor; intro S hS
  apply isLimitOfReflects (sheafToPresheaf J D)
  have : ReflectsLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) (forget D) :=
    reflectsLimitsOfShapeOfReflectsIsomorphisms
  -- porting note: the mathlib proof was by `apply is_limit_of_preserves (J.sheafification D) hS`
  have : PreservesLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K))
      (presheafToSheaf J D ⋙ sheafToPresheaf J D) :=
    preservesLimitsOfShapeOfNatIso (J.sheafificationIsoPresheafToSheafCompSheafToPreasheaf D)
  

Proof state:
case preservesLimit.preserves.t
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁴ : PreservesLimits (forget D)
inst✝³ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝² : SmallCategory K
inst✝¹ : FinCategory K
inst✝ : HasLimitsOfShape K D
e : K ≌ AsSmall (FinCategory.AsType K) := (FinCategory.equivAsType K).symm.trans AsSmall.equiv
this✝² : HasLimitsOfShape (AsSmall (FinCategory.AsType K)) D
this✝¹ : FinCategory (AsSmall (FinCategory.AsType K))
F : AsSmall (FinCategory.AsType K) ⥤ Cᵒᵖ ⥤ D
S : Cone F
hS : IsLimit S
this✝ : ReflectsLimitsOfShape (AsSmall (FinCategory.AsType K)) (forget D)
this : PreservesLimitsOfShape (AsSmall (FinCategory.AsType K)) (presheafToSheaf J D ⋙ sheafToPresheaf J D)
⊢ IsLimit ((sheafToPresheaf J D).mapCone ((presheafToSheaf J D).mapCone S))

Next tactic:
exact isLimitOfPreserves (presheafToSheaf J D ⋙ sheafToPresheaf J D) hS
instance preservesfiniteLimits_presheafToSheaf [HasFiniteLimits D] :
    PreservesFiniteLimits (presheafToSheaf J D) := by
  

Proof state:
C : Type u
inst✝¹¹ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹⁰ : Category.{max v u, w} D
inst✝⁹ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁸ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁷ : ConcreteCategory D
inst✝⁶ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁵ : PreservesLimits (forget D)
inst✝⁴ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝³ : SmallCategory K
inst✝² : FinCategory K
inst✝¹ : HasLimitsOfShape K D
inst✝ : HasFiniteLimits D
⊢ PreservesFiniteLimits (presheafToSheaf J D)

Next tactic:
apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{max v u}
instance preservesfiniteLimits_presheafToSheaf [HasFiniteLimits D] :
    PreservesFiniteLimits (presheafToSheaf J D) := by
  apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{max v u}
  

Proof state:
case h
C : Type u
inst✝¹¹ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹⁰ : Category.{max v u, w} D
inst✝⁹ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁸ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁷ : ConcreteCategory D
inst✝⁶ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁵ : PreservesLimits (forget D)
inst✝⁴ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝³ : SmallCategory K
inst✝² : FinCategory K
inst✝¹ : HasLimitsOfShape K D
inst✝ : HasFiniteLimits D
⊢ (J_1 : Type (max v u)) → {𝒥 : SmallCategory J_1} → FinCategory J_1 → PreservesLimitsOfShape J_1 (presheafToSheaf J D)

Next tactic:
intros
instance preservesfiniteLimits_presheafToSheaf [HasFiniteLimits D] :
    PreservesFiniteLimits (presheafToSheaf J D) := by
  apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{max v u}
  intros
  

Proof state:
case h
C : Type u
inst✝¹¹ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹⁰ : Category.{max v u, w} D
inst✝⁹ :
  ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)
inst✝⁸ : ∀ (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ D
inst✝⁷ : ConcreteCategory D
inst✝⁶ : (X : C) → PreservesColimitsOfShape (GrothendieckTopology.Cover J X)ᵒᵖ (forget D)
inst✝⁵ : PreservesLimits (forget D)
inst✝⁴ : ReflectsIsomorphisms (forget D)
K : Type w'
inst✝³ : SmallCategory K
inst✝² : FinCategory K
inst✝¹ : HasLimitsOfShape K D
inst✝ : HasFiniteLimits D
J✝ : Type (max v u)
𝒥✝ : SmallCategory J✝
x✝ : FinCategory J✝
⊢ PreservesLimitsOfShape J✝ (presheafToSheaf J D)

Next tactic:
infer_instance
@[ext]
theorem OverMorphism.ext {X : T} {U V : Over X} {f g : U ⟶ V} (h : f.left = g.left) : f = g := by
  

Proof state:
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X✝ X : T
U V : Over X
f g : U ⟶ V
h : f.left = g.left
⊢ f = g

Next tactic:
let ⟨_,b,_⟩ := f
@[ext]
theorem OverMorphism.ext {X : T} {U V : Over X} {f g : U ⟶ V} (h : f.left = g.left) : f = g := by
  let ⟨_,b,_⟩ := f
  

Proof state:
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X✝ X : T
U V : Over X
f g : U ⟶ V
left✝ : U.left ⟶ V.left
b : U.right ⟶ V.right
w✝ : (𝟭 T).map left✝ ≫ V.hom = U.hom ≫ (Functor.fromPUnit X).map b
h : (CommaMorphism.mk left✝ b).left = g.left
⊢ CommaMorphism.mk left✝ b = g

Next tactic:
let ⟨_,e,_⟩ := g
@[ext]
theorem OverMorphism.ext {X : T} {U V : Over X} {f g : U ⟶ V} (h : f.left = g.left) : f = g := by
  let ⟨_,b,_⟩ := f
  let ⟨_,e,_⟩ := g
  

Proof state:
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X✝ X : T
U V : Over X
f g : U ⟶ V
left✝¹ : U.left ⟶ V.left
b : U.right ⟶ V.right
w✝¹ : (𝟭 T).map left✝¹ ≫ V.hom = U.hom ≫ (Functor.fromPUnit X).map b
left✝ : U.left ⟶ V.left
e : U.right ⟶ V.right
w✝ : (𝟭 T).map left✝ ≫ V.hom = U.hom ≫ (Functor.fromPUnit X).map e
h : (CommaMorphism.mk left✝¹ b).left = (CommaMorphism.mk left✝ e).left
⊢ CommaMorphism.mk left✝¹ b = CommaMorphism.mk left✝ e

Next tactic:
congr
@[ext]
theorem OverMorphism.ext {X : T} {U V : Over X} {f g : U ⟶ V} (h : f.left = g.left) : f = g := by
  let ⟨_,b,_⟩ := f
  let ⟨_,e,_⟩ := g
  congr
  

Proof state:
case e_right
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X✝ X : T
U V : Over X
f g : U ⟶ V
left✝¹ : U.left ⟶ V.left
b : U.right ⟶ V.right
w✝¹ : (𝟭 T).map left✝¹ ≫ V.hom = U.hom ≫ (Functor.fromPUnit X).map b
left✝ : U.left ⟶ V.left
e : U.right ⟶ V.right
w✝ : (𝟭 T).map left✝ ≫ V.hom = U.hom ≫ (Functor.fromPUnit X).map e
h : (CommaMorphism.mk left✝¹ b).left = (CommaMorphism.mk left✝ e).left
⊢ b = e

Next tactic:
simp only [eq_iff_true_of_subsingleton]
theorem over_right (U : Over X) : U.right = ⟨⟨⟩⟩ := by 

Proof state:
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
U : Over X
⊢ U.right = { as := PUnit.unit }

Next tactic:
simp only
@[reassoc (attr := simp)]
theorem w {A B : Over X} (f : A ⟶ B) : f.left ≫ B.hom = A.hom := by 

Proof state:
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
A B : Over X
f : A ⟶ B
⊢ f.left ≫ B.hom = A.hom

Next tactic:
have := f.w
@[reassoc (attr := simp)]
theorem w {A B : Over X} (f : A ⟶ B) : f.left ≫ B.hom = A.hom := by have := f.w; 

Proof state:
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
A B : Over X
f : A ⟶ B
this : (𝟭 T).map f.left ≫ B.hom = A.hom ≫ (Functor.fromPUnit X).map f.right
⊢ f.left ≫ B.hom = A.hom

Next tactic:
aesop_cat
instance forget_reflects_iso : ReflectsIsomorphisms (forget X) where
  reflects {Y Z} f t := by
    

Proof state:
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
Y Z : Over X
f : Y ⟶ Z
t : IsIso ((forget X).map f)
⊢ IsIso f

Next tactic:
let g : Z ⟶ Y := Over.homMk (inv ((forget X).map f))
      ((asIso ((forget X).map f)).inv_comp_eq.2 (Over.w f).symm)
instance forget_reflects_iso : ReflectsIsomorphisms (forget X) where
  reflects {Y Z} f t := by
    let g : Z ⟶ Y := Over.homMk (inv ((forget X).map f))
      ((asIso ((forget X).map f)).inv_comp_eq.2 (Over.w f).symm)
    

Proof state:
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
Y Z : Over X
f : Y ⟶ Z
t : IsIso ((forget X).map f)
g : Z ⟶ Y := homMk (inv ((forget X).map f))
⊢ IsIso f

Next tactic:
dsimp [forget] at t
instance forget_reflects_iso : ReflectsIsomorphisms (forget X) where
  reflects {Y Z} f t := by
    let g : Z ⟶ Y := Over.homMk (inv ((forget X).map f))
      ((asIso ((forget X).map f)).inv_comp_eq.2 (Over.w f).symm)
    dsimp [forget] at t
    

Proof state:
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
Y Z : Over X
f : Y ⟶ Z
t : IsIso f.left
g : Z ⟶ Y := homMk (inv ((forget X).map f))
⊢ IsIso f

Next tactic:
refine ⟨⟨g, ⟨?_,?_⟩⟩⟩
instance forget_reflects_iso : ReflectsIsomorphisms (forget X) where
  reflects {Y Z} f t := by
    let g : Z ⟶ Y := Over.homMk (inv ((forget X).map f))
      ((asIso ((forget X).map f)).inv_comp_eq.2 (Over.w f).symm)
    dsimp [forget] at t
    refine ⟨⟨g, ⟨?_,?_⟩⟩⟩
    

Proof state:
case refine_1
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
Y Z : Over X
f : Y ⟶ Z
t : IsIso f.left
g : Z ⟶ Y := homMk (inv ((forget X).map f))
⊢ f ≫ g = 𝟙 Y
case refine_2
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
Y Z : Over X
f : Y ⟶ Z
t : IsIso f.left
g : Z ⟶ Y := homMk (inv ((forget X).map f))
⊢ g ≫ f = 𝟙 Z

Next tactic:
repeat (ext; simp)
instance forget_reflects_iso : ReflectsIsomorphisms (forget X) where
  reflects {Y Z} f t := by
    let g : Z ⟶ Y := Over.homMk (inv ((forget X).map f))
      ((asIso ((forget X).map f)).inv_comp_eq.2 (Over.w f).symm)
    dsimp [forget] at t
    refine ⟨⟨g, ⟨?_,?_⟩⟩⟩
    repeat (

Proof state:
case refine_1
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
Y Z : Over X
f : Y ⟶ Z
t : IsIso f.left
g : Z ⟶ Y := homMk (inv ((forget X).map f))
⊢ f ≫ g = 𝟙 Y
case refine_2
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
Y Z : Over X
f : Y ⟶ Z
t : IsIso f.left
g : Z ⟶ Y := homMk (inv ((forget X).map f))
⊢ g ≫ f = 𝟙 Z

Next tactic:
ext