l3lab/ntp-mathlib huggingface dataset:
instance preservesLimitsOfShape_plusFunctor
    (K : Type max v u) [SmallCategory K] [FinCategory K] [HasLimitsOfShape K D]
    [PreservesLimitsOfShape K (forget D)] [ReflectsLimitsOfShape K (forget D)] :
    PreservesLimitsOfShape K (J.plusFunctor D) := by
  constructor; intro F; apply preservesLimitOfEvaluation; intro X
  apply preservesLimitOfPreservesLimitCone (limit.isLimit F)
  refine' ⟨fun S => liftToPlusObjLimitObj.{w, v, u} F X.unop S, _, _⟩
  · intro S k
    apply liftToPlusObjLimitObj_fac
  · intro S m hm
    dsimp [liftToPlusObjLimitObj]
    simp_rw [← Category.assoc, Iso.eq_comp_inv, ← Iso.comp_inv_eq]
    refine' limit.hom_ext (fun k => _)
    simp only [limit.lift_π, Category.assoc, ← hm]
    congr 1
    refine' colimit.hom_ext (fun k => _)
    dsimp [plusMap, plusObj]
    erw [colimit.ι_map, colimit.ι_desc_assoc, limit.lift_π]
    conv_lhs => dsimp
    simp only [Category.assoc]
    rw [ι_colimitLimitIso_limit_π_assoc]
    simp only [NatIso.ofComponents_inv_app, colimitObjIsoColimitCompEvaluation_ι_app_hom,
      Iso.symm_inv]
    

Proof state:
case preservesLimit.H.refine'_2.e_a
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 : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : Cᵒᵖ
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X)
m : S.pt ⟶ ((plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).pt
hm : ∀ (j : K), m ≫ ((plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).π.app j = S.π.app j
k✝ : K
k : (Cover J X.unop)ᵒᵖ
⊢ (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))
              (limit.isLimit (F ⋙ diagramFunctor J D X.unop))).hom.app
        k ≫
      (limit.π (F ⋙ diagramFunctor J D X.unop) k✝).app k ≫
        colimit.ι (Functor.flip (F ⋙ diagramFunctor J D X.unop) ⋙ (evaluation K D).obj k✝) k =
    (diagramNatTrans J (limit.π F k✝) X.unop).app k ≫ colimit.ι (diagram J (F.obj k✝) X.unop) k

Next tactic:
conv_lhs =>
      dsimp [IsLimit.conePointUniqueUpToIso]
instance preservesLimitsOfShape_plusFunctor
    (K : Type max v u) [SmallCategory K] [FinCategory K] [HasLimitsOfShape K D]
    [PreservesLimitsOfShape K (forget D)] [ReflectsLimitsOfShape K (forget D)] :
    PreservesLimitsOfShape K (J.plusFunctor D) := by
  constructor; intro F; apply preservesLimitOfEvaluation; intro X
  apply preservesLimitOfPreservesLimitCone (limit.isLimit F)
  refine' ⟨fun S => liftToPlusObjLimitObj.{w, v, u} F X.unop S, _, _⟩
  · intro S k
    apply liftToPlusObjLimitObj_fac
  · intro S m hm
    dsimp [liftToPlusObjLimitObj]
    simp_rw [← Category.assoc, Iso.eq_comp_inv, ← Iso.comp_inv_eq]
    refine' limit.hom_ext (fun k => _)
    simp only [limit.lift_π, Category.assoc, ← hm]
    congr 1
    refine' colimit.hom_ext (fun k => _)
    dsimp [plusMap, plusObj]
    erw [colimit.ι_map, colimit.ι_desc_assoc, limit.lift_π]
    conv_lhs => dsimp
    simp only [Category.assoc]
    rw [ι_colimitLimitIso_limit_π_assoc]
    simp only [NatIso.ofComponents_inv_app, colimitObjIsoColimitCompEvaluation_ι_app_hom,
      Iso.symm_inv]
    conv_lhs =>
      

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 : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : Cᵒᵖ
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X)
m : S.pt ⟶ ((plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).pt
hm : ∀ (j : K), m ≫ ((plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).π.app j = S.π.app j
k✝ : K
k : (Cover J X.unop)ᵒᵖ
| (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))
            (limit.isLimit (F ⋙ diagramFunctor J D X.unop))).hom.app
      k ≫
    (limit.π (F ⋙ diagramFunctor J D X.unop) k✝).app k ≫
      colimit.ι (Functor.flip (F ⋙ diagramFunctor J D X.unop) ⋙ (evaluation K D).obj k✝) k

Next tactic:
dsimp [IsLimit.conePointUniqueUpToIso]
instance preservesLimitsOfShape_plusFunctor
    (K : Type max v u) [SmallCategory K] [FinCategory K] [HasLimitsOfShape K D]
    [PreservesLimitsOfShape K (forget D)] [ReflectsLimitsOfShape K (forget D)] :
    PreservesLimitsOfShape K (J.plusFunctor D) := by
  constructor; intro F; apply preservesLimitOfEvaluation; intro X
  apply preservesLimitOfPreservesLimitCone (limit.isLimit F)
  refine' ⟨fun S => liftToPlusObjLimitObj.{w, v, u} F X.unop S, _, _⟩
  · intro S k
    apply liftToPlusObjLimitObj_fac
  · intro S m hm
    dsimp [liftToPlusObjLimitObj]
    simp_rw [← Category.assoc, Iso.eq_comp_inv, ← Iso.comp_inv_eq]
    refine' limit.hom_ext (fun k => _)
    simp only [limit.lift_π, Category.assoc, ← hm]
    congr 1
    refine' colimit.hom_ext (fun k => _)
    dsimp [plusMap, plusObj]
    erw [colimit.ι_map, colimit.ι_desc_assoc, limit.lift_π]
    conv_lhs => dsimp
    simp only [Category.assoc]
    rw [ι_colimitLimitIso_limit_π_assoc]
    simp only [NatIso.ofComponents_inv_app, colimitObjIsoColimitCompEvaluation_ι_app_hom,
      Iso.symm_inv]
    conv_lhs =>
      dsimp [IsLimit.conePointUniqueUpToIso]
    

Proof state:
case preservesLimit.H.refine'_2.e_a
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 : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : Cᵒᵖ
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X)
m : S.pt ⟶ ((plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).pt
hm : ∀ (j : K), m ≫ ((plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).π.app j = S.π.app j
k✝ : K
k : (Cover J X.unop)ᵒᵖ
⊢ (limit.lift (F ⋙ diagramFunctor J D X.unop) ((diagramFunctor J D X.unop).mapCone (limit.cone F))).app k ≫
      (limit.π (F ⋙ diagramFunctor J D X.unop) k✝).app k ≫
        colimit.ι (Functor.flip (F ⋙ diagramFunctor J D X.unop) ⋙ (evaluation K D).obj k✝) k =
    (diagramNatTrans J (limit.π F k✝) X.unop).app k ≫ colimit.ι (diagram J (F.obj k✝) X.unop) k

Next tactic:
rw [← Category.assoc, ← NatTrans.comp_app, limit.lift_π]
instance preservesLimitsOfShape_plusFunctor
    (K : Type max v u) [SmallCategory K] [FinCategory K] [HasLimitsOfShape K D]
    [PreservesLimitsOfShape K (forget D)] [ReflectsLimitsOfShape K (forget D)] :
    PreservesLimitsOfShape K (J.plusFunctor D) := by
  constructor; intro F; apply preservesLimitOfEvaluation; intro X
  apply preservesLimitOfPreservesLimitCone (limit.isLimit F)
  refine' ⟨fun S => liftToPlusObjLimitObj.{w, v, u} F X.unop S, _, _⟩
  · intro S k
    apply liftToPlusObjLimitObj_fac
  · intro S m hm
    dsimp [liftToPlusObjLimitObj]
    simp_rw [← Category.assoc, Iso.eq_comp_inv, ← Iso.comp_inv_eq]
    refine' limit.hom_ext (fun k => _)
    simp only [limit.lift_π, Category.assoc, ← hm]
    congr 1
    refine' colimit.hom_ext (fun k => _)
    dsimp [plusMap, plusObj]
    erw [colimit.ι_map, colimit.ι_desc_assoc, limit.lift_π]
    conv_lhs => dsimp
    simp only [Category.assoc]
    rw [ι_colimitLimitIso_limit_π_assoc]
    simp only [NatIso.ofComponents_inv_app, colimitObjIsoColimitCompEvaluation_ι_app_hom,
      Iso.symm_inv]
    conv_lhs =>
      dsimp [IsLimit.conePointUniqueUpToIso]
    rw [← Category.assoc, ← NatTrans.comp_app, limit.lift_π]
    

Proof state:
case preservesLimit.H.refine'_2.e_a
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 : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : Cᵒᵖ
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X)
m : S.pt ⟶ ((plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).pt
hm : ∀ (j : K), m ≫ ((plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).π.app j = S.π.app j
k✝ : K
k : (Cover J X.unop)ᵒᵖ
⊢ (((diagramFunctor J D X.unop).mapCone (limit.cone F)).π.app k✝).app k ≫
      colimit.ι (Functor.flip (F ⋙ diagramFunctor J D X.unop) ⋙ (evaluation K D).obj k✝) k =
    (diagramNatTrans J (limit.π F k✝) X.unop).app k ≫ colimit.ι (diagram J (F.obj k✝) X.unop) k

Next tactic:
rfl
instance preserveFiniteLimits_plusFunctor
    [HasFiniteLimits D] [PreservesFiniteLimits (forget D)] [ReflectsIsomorphisms (forget D)] :
    PreservesFiniteLimits (J.plusFunctor D) := by
  

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

Next tactic:
apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{max v u}
instance preserveFiniteLimits_plusFunctor
    [HasFiniteLimits D] [PreservesFiniteLimits (forget D)] [ReflectsIsomorphisms (forget D)] :
    PreservesFiniteLimits (J.plusFunctor D) := by
  apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{max v u}
  

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

Next tactic:
intro K _ _
instance preserveFiniteLimits_plusFunctor
    [HasFiniteLimits D] [PreservesFiniteLimits (forget D)] [ReflectsIsomorphisms (forget D)] :
    PreservesFiniteLimits (J.plusFunctor D) := by
  apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{max v u}
  intro K _ _
  

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

Next tactic:
have : ReflectsLimitsOfShape K (forget D) := reflectsLimitsOfShapeOfReflectsIsomorphisms
instance preserveFiniteLimits_plusFunctor
    [HasFiniteLimits D] [PreservesFiniteLimits (forget D)] [ReflectsIsomorphisms (forget D)] :
    PreservesFiniteLimits (J.plusFunctor D) := by
  apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{max v u}
  intro K _ _
  have : ReflectsLimitsOfShape K (forget D) := reflectsLimitsOfShapeOfReflectsIsomorphisms
  

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

Next tactic:
apply preservesLimitsOfShape_plusFunctor.{w, v, u}
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  

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
⊢ PreservesLimitsOfShape K (presheafToSheaf J D)

Next tactic:
let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
instance preservesLimitsOfShape_presheafToSheaf :
    PreservesLimitsOfShape K (presheafToSheaf J D) := by
  let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, max v u})
  

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
⊢ PreservesLimitsOfShape K (presheafToSheaf J D)

Next tactic:
haveI : HasLimitsOfShape (AsSmall.{max v u} (FinCategory.AsType K)) D :=
    Limits.hasLimitsOfShape_of_equivalence e
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
  

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
⊢ PreservesLimitsOfShape K (presheafToSheaf J D)

Next tactic:
haveI : FinCategory (AsSmall.{max v u} (FinCategory.AsType K)) := by
    constructor
    · show Fintype (ULift _)
      infer_instance
    · intro j j'
      show Fintype (ULift _)
      infer_instance
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
    

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
⊢ FinCategory (AsSmall (FinCategory.AsType K))

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
    · 

Proof state:
case fintypeObj
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
⊢ autoParam (Fintype (AsSmall (FinCategory.AsType K))) _auto✝

Next tactic:
show Fintype (ULift _)
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 _)
      

Proof state:
case fintypeObj
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
⊢ Fintype (ULift.{max u v, 0} (FinCategory.AsType K))

Next tactic:
infer_instance
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
    · 

Proof state:
case fintypeHom
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
⊢ autoParam ((j j' : AsSmall (FinCategory.AsType K)) → Fintype (j ⟶ j')) _auto✝

Next tactic:
intro j j'
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'
      

Proof state:
case fintypeHom
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
j j' : AsSmall (FinCategory.AsType K)
⊢ Fintype (j ⟶ j')

Next tactic:
show Fintype (ULift _)
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 _)
      

Proof state:
case fintypeHom
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
j j' : AsSmall (FinCategory.AsType K)
⊢ Fintype (ULift.{max u v, 0} (j.down ⟶ j'.down))

Next tactic:
infer_instance