awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Limits.Indization.Category
Goal:
C : Type u
inst : Category.{v, u} C
A B : Ind.{v, u} C
f : A ⟶ B
obtain :
  ∀
    (P :
      IndParallelPairPresentation (A := A.obj (P := IsIndObject)) (B := B.obj (P := IsIndObject))
        (map (Ind.inclusion C) f : obj (Ind.inclusion C) A ⟶ obj (Ind.inclusion C) B) (map (Ind.inclusion C) f)),
    ∃ (I : Type v) (x : SmallCategory I) (x_1 : IsFiltered I) (F : I ⥤ C) (G : I ⥤ C) (φ : F ⟶ G),
      Nonempty.{v + 1} (Arrow.mk f ≅ Arrow.mk (X := obj (Ind.lim I) F) (map (Ind.lim I) φ))
⊢ ∃ (I : Type v) (x : SmallCategory I) (x_1 : IsFiltered I) (F : I ⥤ C) (G : I ⥤ C) (φ : F ⟶ G),
    Nonempty.{v + 1} (Arrow.mk f ≅ Arrow.mk (X := obj (Ind.lim I) F) (map (Ind.lim I) φ))

Tactic:
obtain ⟨P⟩ := nonempty_indParallelPairPresentation A.2 B.2
    (Ind.inclusion _ |>.map f) (Ind.inclusion _ |>.map f)

Premises used: CategoryTheory.ObjectProperty.FullSubcategory.property, CategoryTheory.Ind.inclusion, CategoryTheory.Functor.map, CategoryTheory.nonempty_indParallelPairPresentation
-- Module: Mathlib.CategoryTheory.Limits.Indization.Category
Goal:
C : Type u
inst : Category.{v, u} C
A B : Ind.{v, u} C
f : A ⟶ B
P :
  IndParallelPairPresentation (A := A.obj (P := IsIndObject)) (B := B.obj (P := IsIndObject))
    (map (Ind.inclusion C) f : obj (Ind.inclusion C) A ⟶ obj (Ind.inclusion C) B) (map (Ind.inclusion C) f)
⊢ ∃ (I : Type v) (x : SmallCategory I) (x_1 : IsFiltered I) (F : I ⥤ C) (G : I ⥤ C) (φ : F ⟶ G),
    Nonempty.{v + 1} (Arrow.mk f ≅ Arrow.mk (X := obj (Ind.lim I) F) (map (Ind.lim I) φ))

Tactic:
⟨P.I, inferInstance, inferInstance, P.F₁, P.F₂, P.φ, ⟨Arrow.isoMk ?_ ?_ ?_⟩⟩

Premises used: CategoryTheory.IndParallelPairPresentation.φ, CategoryTheory.IndParallelPairPresentation.F₂, inferInstance, Nonempty.intro, Exists.intro, CategoryTheory.IndParallelPairPresentation.I, CategoryTheory.IndParallelPairPresentation.F₁, CategoryTheory.Arrow.isoMk
-- Module: Mathlib.CategoryTheory.Limits.Indization.Category
Goal:
C : Type u
inst : Category.{v, u} C
A B : Ind.{v, u} C
f : A ⟶ B
P :
  IndParallelPairPresentation (A := A.obj (P := IsIndObject)) (B := B.obj (P := IsIndObject))
    (map (Ind.inclusion C) f : obj (Ind.inclusion C) A ⟶ obj (Ind.inclusion C) B) (map (Ind.inclusion C) f)
refine_1 : (Arrow.mk f).left ≅ (Arrow.mk (X := obj (Ind.lim P.I) P.F₁) (map (Ind.lim P.I) P.φ)).left
refine_2 : (Arrow.mk f).right ≅ (Arrow.mk (X := obj (Ind.lim P.I) P.F₁) (map (Ind.lim P.I) P.φ)).right
refine_3 :
  (P.parallelPairIsoParallelPairCompIndYoneda.app WalkingParallelPair.zero).hom ≫
      (Arrow.mk (X := obj (Ind.lim P.I) P.F₁) (map (Ind.lim P.I) P.φ)).hom =
    (Arrow.mk f).hom ≫ (P.parallelPairIsoParallelPairCompIndYoneda.app WalkingParallelPair.one).hom
⊢ ∃ (I : Type v) (x : SmallCategory I) (x_1 : IsFiltered I) (F : I ⥤ C) (G : I ⥤ C) (φ : F ⟶ G),
    Nonempty.{v + 1} (Arrow.mk f ≅ Arrow.mk (X := obj (Ind.lim I) F) (map (Ind.lim I) φ))

Tactic:
refine ⟨P.I, inferInstance, inferInstance, P.F₁, P.F₂, P.φ, ⟨Arrow.isoMk ?_ ?_ ?_⟩⟩
-- Module: Mathlib.CategoryTheory.Limits.Indization.Category
Goal:
C : Type u
inst : Category.{v, u} C
A B : Ind.{v, u} C
f : A ⟶ B
P :
  IndParallelPairPresentation (A := A.obj (P := IsIndObject)) (B := B.obj (P := IsIndObject))
    (map (Ind.inclusion C) f : obj (Ind.inclusion C) A ⟶ obj (Ind.inclusion C) B) (map (Ind.inclusion C) f)
⊢ (Arrow.mk f).left ≅ (Arrow.mk (X := obj (Ind.lim P.I) P.F₁) (map (Ind.lim P.I) P.φ)).left

Tactic:
exact P.parallelPairIsoParallelPairCompIndYoneda.app WalkingParallelPair.zero

Premises used: CategoryTheory.IndParallelPairPresentation.parallelPairIsoParallelPairCompIndYoneda, CategoryTheory.Iso.app, CategoryTheory.Limits.WalkingParallelPair.zero
-- Module: Mathlib.CategoryTheory.Limits.Indization.Category
Goal:
C : Type u
inst : Category.{v, u} C
A B : Ind.{v, u} C
f : A ⟶ B
P :
  IndParallelPairPresentation (A := A.obj (P := IsIndObject)) (B := B.obj (P := IsIndObject))
    (map (Ind.inclusion C) f : obj (Ind.inclusion C) A ⟶ obj (Ind.inclusion C) B) (map (Ind.inclusion C) f)
⊢ (Arrow.mk f).right ≅ (Arrow.mk (X := obj (Ind.lim P.I) P.F₁) (map (Ind.lim P.I) P.φ)).right

Tactic:
exact P.parallelPairIsoParallelPairCompIndYoneda.app WalkingParallelPair.one

Premises used: CategoryTheory.Limits.WalkingParallelPair.one, CategoryTheory.IndParallelPairPresentation.parallelPairIsoParallelPairCompIndYoneda, CategoryTheory.Iso.app
-- Module: Mathlib.CategoryTheory.Limits.Indization.Category
Goal:
C : Type u
inst : Category.{v, u} C
A B : Ind.{v, u} C
f : A ⟶ B
P :
  IndParallelPairPresentation (A := A.obj (P := IsIndObject)) (B := B.obj (P := IsIndObject))
    (map (Ind.inclusion C) f : obj (Ind.inclusion C) A ⟶ obj (Ind.inclusion C) B) (map (Ind.inclusion C) f)
⊢ (P.parallelPairIsoParallelPairCompIndYoneda.app WalkingParallelPair.zero).hom ≫
      (Arrow.mk (X := obj (Ind.lim P.I) P.F₁) (map (Ind.lim P.I) P.φ)).hom =
    (Arrow.mk f).hom ≫ (P.parallelPairIsoParallelPairCompIndYoneda.app WalkingParallelPair.one).hom

Tactic:
simpa using
      (P.parallelPairIsoParallelPairCompIndYoneda.hom.naturality WalkingParallelPairHom.left).symm

Premises used: CategoryTheory.IndParallelPairPresentation.parallelPairIsoParallelPairCompIndYoneda, CategoryTheory.NatTrans.naturality, Eq.symm, CategoryTheory.Iso.hom, CategoryTheory.Limits.WalkingParallelPairHom.left
-- Module: Mathlib.CategoryTheory.Limits.Indization.Category
Goal:
C_1 : Type u
inst : Category.{v, u} C_1
C : Type u
inst_1 : SmallCategory C
inst_2 : HasFiniteColimits C
h : ∀ (x : Cᵒᵖ ⥤ Type u), IsIndObject x ↔ leftExactFunctor Cᵒᵖ (Type u) x
⊢ Eq (α := ObjectProperty (Cᵒᵖ ⥤ Type u)) IsIndObject (leftExactFunctor Cᵒᵖ (Type u))

Tactic:
ext
-- Module: Mathlib.CategoryTheory.Limits.Indization.Category
Goal:
C_1 : Type u
inst : Category.{v, u} C_1
C : Type u
inst_1 : SmallCategory C
inst_2 : HasFiniteColimits C
x : Cᵒᵖ ⥤ Type u
⊢ IsIndObject x ↔ leftExactFunctor Cᵒᵖ (Type u) x

Tactic:
apply isIndObject_iff_preservesFiniteLimits

Premises used: CategoryTheory.Limits.isIndObject_iff_preservesFiniteLimits
-- Module: Mathlib.CategoryTheory.Limits.Indization.Equalizers
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
J : Type
inst_3 : SmallCategory J
inst_4 : FinCategory J
F : J ⥤ I ⥤ C
hF : ∀ (i : I), IsIndObject (limit (F.flip.obj i ⋙ yoneda))
G : J ⥤ I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v) :=
  F ⋙ Functor.obj (Functor.whiskeringRight I C (Functor.{v, v, u, v + 1} Cᵒᵖ (Type v))) yoneda
apply : IsIndObject (limit (colimit G.flip))
⊢ IsIndObject
    (limit (F ⋙ Functor.obj (Functor.whiskeringRight I C (Functor.{v, v, u, v + 1} Cᵒᵖ (Type v))) yoneda ⋙ colim))

Tactic:
apply IsIndObject.map (HasLimit.isoOfNatIso (colimitFlipIsoCompColim G)).hom

Premises used: CategoryTheory.Limits.HasLimit.isoOfNatIso, CategoryTheory.Limits.IsIndObject.map, CategoryTheory.Limits.colimitFlipIsoCompColim, CategoryTheory.Iso.hom
-- Module: Mathlib.CategoryTheory.Limits.Indization.Equalizers
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
J : Type
inst_3 : SmallCategory J
inst_4 : FinCategory J
F : J ⥤ I ⥤ C
hF : ∀ (i : I), IsIndObject (limit (F.flip.obj i ⋙ yoneda))
G : J ⥤ I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v) :=
  F ⋙ Functor.obj (Functor.whiskeringRight I C (Functor.{v, v, u, v + 1} Cᵒᵖ (Type v))) yoneda
apply : IsIndObject (colimit (limit G))
⊢ IsIndObject (limit (colimit G.flip))

Tactic:
apply IsIndObject.map (colimitLimitIso G).hom

Premises used: CategoryTheory.Limits.colimitLimitIso, CategoryTheory.Limits.IsIndObject.map, CategoryTheory.Iso.hom
-- Module: Mathlib.CategoryTheory.Limits.Indization.Equalizers
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
J : Type
inst_3 : SmallCategory J
inst_4 : FinCategory J
F : J ⥤ I ⥤ C
hF : ∀ (i : I), IsIndObject (limit (F.flip.obj i ⋙ yoneda))
G : J ⥤ I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v) :=
  F ⋙ Functor.obj (Functor.whiskeringRight I C (Functor.{v, v, u, v + 1} Cᵒᵖ (Type v))) yoneda
hF₁ : ∀ (i : I), IsIndObject ((limit G).obj i)
⊢ IsIndObject (colimit (limit G))

Tactic:
apply isIndObject_colimit

Premises used: CategoryTheory.Limits.isIndObject_colimit
-- Module: Mathlib.CategoryTheory.Limits.Indization.Equalizers
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
J : Type
inst_3 : SmallCategory J
inst_4 : FinCategory J
F : J ⥤ I ⥤ C
hF : ∀ (i : I), IsIndObject (limit (F.flip.obj i ⋙ yoneda))
G : J ⥤ I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v) :=
  F ⋙ Functor.obj (Functor.whiskeringRight I C (Functor.{v, v, u, v + 1} Cᵒᵖ (Type v))) yoneda
⊢ ∀ (i : I), IsIndObject ((limit G).obj i)

Tactic:
fun i => IsIndObject.map (limitObjIsoLimitCompEvaluation _ _).inv (hF i)

Premises used: CategoryTheory.Limits.IsIndObject.map, CategoryTheory.Limits.limitObjIsoLimitCompEvaluation, CategoryTheory.Iso.inv
-- Module: Mathlib.CategoryTheory.Limits.Indization.Equalizers
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
J : Type
inst_3 : SmallCategory J
inst_4 : FinCategory J
F : J ⥤ I ⥤ C
hF : ∀ (i : I), IsIndObject (limit (F.flip.obj i ⋙ yoneda))
G : J ⥤ I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v) :=
  F ⋙ Functor.obj (Functor.whiskeringRight I C (Functor.{v, v, u, v + 1} Cᵒᵖ (Type v))) yoneda
⊢ ∀ (i : I), IsIndObject ((limit G).obj i)

Tactic:
exact fun i => IsIndObject.map (limitObjIsoLimitCompEvaluation _ _).inv (hF i)

Premises used: CategoryTheory.Limits.IsIndObject.map, CategoryTheory.Limits.limitObjIsoLimitCompEvaluation, CategoryTheory.Iso.inv
-- Module: Mathlib.CategoryTheory.Limits.Indization.Equalizers
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasEqualizers.{v, u} C
F : WalkingParallelPair ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : HasLimit F
h : ∀ (j : WalkingParallelPair), IsIndObject (F.obj j)
obtain :
  ∀ (P : IndParallelPairPresentation (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right)),
    IsIndObject (limit F)
⊢ IsIndObject (limit F)

Tactic:
obtain ⟨P⟩ := nonempty_indParallelPairPresentation (h WalkingParallelPair.zero)
      (h WalkingParallelPair.one) (F.map WalkingParallelPairHom.left)
      (F.map WalkingParallelPairHom.right)

Premises used: CategoryTheory.Limits.WalkingParallelPair.one, CategoryTheory.Functor.map, CategoryTheory.Limits.WalkingParallelPairHom.right, CategoryTheory.nonempty_indParallelPairPresentation, CategoryTheory.Limits.WalkingParallelPairHom.left, CategoryTheory.Limits.WalkingParallelPair.zero
-- Module: Mathlib.CategoryTheory.Limits.Indization.Equalizers
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasEqualizers.{v, u} C
F : WalkingParallelPair ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : HasLimit F
h : ∀ (j : WalkingParallelPair), IsIndObject (F.obj j)
P : IndParallelPairPresentation (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right)
⊢ IsIndObject (limit F)

Tactic:
IsIndObject.map
      (HasLimit.isoOfNatIso (P.parallelPairIsoParallelPairCompYoneda.symm ≪≫
        (diagramIsoParallelPair _).symm)).hom
      (isIndObject_limit_comp_yoneda_comp_colim (parallelPair P.φ P.ψ)
        (fun i => isIndObject_limit_comp_yoneda _))

Premises used: CategoryTheory.Limits.HasLimit.isoOfNatIso, CategoryTheory.Functor.flip, CategoryTheory.IndParallelPairPresentation.φ, CategoryTheory.IndParallelPairPresentation.parallelPairIsoParallelPairCompYoneda, CategoryTheory.Limits.IsIndObject.map, CategoryTheory.IndParallelPairPresentation.ψ, CategoryTheory.Limits.diagramIsoParallelPair, CategoryTheory.Limits.isIndObject_limit_comp_yoneda, CategoryTheory.Iso.trans, CategoryTheory.Iso.symm, CategoryTheory.Limits.isIndObject_limit_comp_yoneda_comp_colim, CategoryTheory.Iso.hom, CategoryTheory.Limits.parallelPair, CategoryTheory.Functor.obj
-- Module: Mathlib.CategoryTheory.Limits.Indization.Equalizers
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasEqualizers.{v, u} C
F : WalkingParallelPair ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : HasLimit F
h : ∀ (j : WalkingParallelPair), IsIndObject (F.obj j)
P : IndParallelPairPresentation (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right)
⊢ IsIndObject (limit F)

Tactic:
exact IsIndObject.map
      (HasLimit.isoOfNatIso (P.parallelPairIsoParallelPairCompYoneda.symm ≪≫
        (diagramIsoParallelPair _).symm)).hom
      (isIndObject_limit_comp_yoneda_comp_colim (parallelPair P.φ P.ψ)
        (fun i => isIndObject_limit_comp_yoneda _))

Premises used: CategoryTheory.Limits.HasLimit.isoOfNatIso, CategoryTheory.Functor.flip, CategoryTheory.IndParallelPairPresentation.φ, CategoryTheory.IndParallelPairPresentation.parallelPairIsoParallelPairCompYoneda, CategoryTheory.Limits.IsIndObject.map, CategoryTheory.IndParallelPairPresentation.ψ, CategoryTheory.Limits.diagramIsoParallelPair, CategoryTheory.Limits.isIndObject_limit_comp_yoneda, CategoryTheory.Iso.trans, CategoryTheory.Iso.symm, CategoryTheory.Limits.isIndObject_limit_comp_yoneda_comp_colim, CategoryTheory.Iso.hom, CategoryTheory.Limits.parallelPair, CategoryTheory.Functor.obj
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
J : Type v
inst_2 : SmallCategory J
inst_3 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst_4 : SmallCategory K
H : K ⥤ Over (colimit F)
inst_5 : IsFiltered K
h : Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (colimit H)))
obtain :
  ∀ (t : limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (colimit H))),
    ∃ (k : K),
      Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (H.obj k)))
⊢ ∃ (k : K), Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (H.obj k)))

Tactic:
obtain ⟨t⟩ := h
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
J : Type v
inst_2 : SmallCategory J
inst_3 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst_4 : SmallCategory K
H : K ⥤ Over (colimit F)
inst_5 : IsFiltered K
c : Cocone H
hc : IsColimit c
T : Over (colimit F)
hT : c.pt ≅ T
h : Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda T))
⊢ ∃ (k : K), Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (H.obj k)))

Tactic:
exists_nonempty_limit_obj_of_colimit F G H ?_

Premises used: CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.exists_nonempty_limit_obj_of_colimit
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
J : Type v
inst_2 : SmallCategory J
inst_3 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst_4 : SmallCategory K
H : K ⥤ Over (colimit F)
inst_5 : IsFiltered K
c : Cocone H
hc : IsColimit c
T : Over (colimit F)
hT : c.pt ≅ T
h : Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda T))
refine : Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (colimit H)))
⊢ ∃ (k : K), Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (H.obj k)))

Tactic:
refine exists_nonempty_limit_obj_of_colimit F G H ?_

Premises used: CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.exists_nonempty_limit_obj_of_colimit
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
J : Type v
inst_2 : SmallCategory J
inst_3 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst_4 : SmallCategory K
H : K ⥤ Over (colimit F)
inst_5 : IsFiltered K
c : Cocone H
hc : IsColimit c
T : Over (colimit F)
hT : c.pt ≅ T
h : Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda T))
suffices : T ≅ colimit H
⊢ Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (colimit H)))

Tactic:
suffices T ≅ colimit H from Nonempty.map (lim.map (whiskerLeft 𝒢 (yoneda.map this.hom))) h
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
J : Type v
inst_2 : SmallCategory J
inst_3 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst_4 : SmallCategory K
H : K ⥤ Over (colimit F)
inst_5 : IsFiltered K
c : Cocone H
hc : IsColimit c
T : Over (colimit F)
hT : c.pt ≅ T
h : Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda T))
⊢ T ≅ colimit H

Tactic:
refine hT.symm ≪≫ IsColimit.coconePointUniqueUpToIso hc (colimit.isColimit _)

Premises used: CategoryTheory.Limits.colimit.isColimit, CategoryTheory.Iso.symm, CategoryTheory.Limits.IsColimit.coconePointUniqueUpToIso
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
⊢ IsFiltered (CostructuredArrow yoneda (colimit F))

Tactic:
IsFiltered.iff_nonempty_limit.mpr (fun {J _ _} G => ?_)

Premises used: Iff.mpr, CategoryTheory.IsFiltered.iff_nonempty_limit
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
refine :
  ∀ {J : Type v} {x : SmallCategory J} {x_1 : FinCategory J} (G : J ⥤ CostructuredArrow yoneda (colimit F)),
    ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
⊢ IsFiltered (CostructuredArrow yoneda (colimit F))

Tactic:
refine IsFiltered.iff_nonempty_limit.mpr (fun {J _ _} G => ?_)

Premises used: Iff.mpr, CategoryTheory.IsFiltered.iff_nonempty_limit
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
⊢ ∀ (j j' : Jᵒᵖ) (f : j ⟶ j'),
    Eq (α :=
      (fun (X : Type (max u v)) ↦ X)
        ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))).obj
          j'))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            ((G.op ⋙
                  Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))).map
              f) :
          (a :
              (fun (X : Type (max u v)) ↦ X)
                ((G.op ⋙ (CostructuredArrow.toOver yoneda (colimit F)).op ⋙ yoneda.obj (Over.mk (𝟙 (colimit F)))).obj
                  j)) →
            (fun (X : Type (max u v)) ↦ X)
              ((G.op ⋙ (CostructuredArrow.toOver yoneda (colimit F)).op ⋙ yoneda.obj (Over.mk (𝟙 (colimit F)))).obj j'))
        ((fun (j : Jᵒᵖ) ↦
            IsTerminal.from Over.mkIdTerminal
              (Opposite.unop (obj (Functor.op (CostructuredArrow.toOver yoneda (colimit F))) (G.op.obj j))))
          j))
      ((fun (j : Jᵒᵖ) ↦
          IsTerminal.from Over.mkIdTerminal
            (Opposite.unop (obj (Functor.op (CostructuredArrow.toOver yoneda (colimit F))) (G.op.obj j))))
        j')

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
⊢ Nonempty
    (limit (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))))

Tactic:
have h₁ : Nonempty (limit (G.op ⋙ (CostructuredArrow.toOver _ _).op ⋙
      yoneda.obj (Over.mk (𝟙 (colimit F))))) :=
    ⟨Types.Limit.mk _ (fun j => Over.mkIdTerminal.from _) (by simp)⟩
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ :
  Nonempty
    (limit (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))))
obtain :
  ∀ (i : I)
    (hi :
      Nonempty
        (limit
          ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙
            obj yoneda (obj (colimit.toCostructuredArrow F ⋙ CostructuredArrow.toOver F (colimit F)) i)))),
    ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
⊢ ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))

Tactic:
obtain ⟨i, hi⟩ := exists_nonempty_limit_obj_of_isColimit F G _
    (colimit.isColimitToOver F) _ (Iso.refl _) h₁

Premises used: CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.exists_nonempty_limit_obj_of_isColimit, CategoryTheory.Limits.colimit.isColimitToOver, CategoryTheory.Iso.refl
-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ :
  Nonempty
    (limit (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙
        obj yoneda (obj (colimit.toCostructuredArrow F ⋙ CostructuredArrow.toOver F (colimit F)) i)))
obtain :
  ∀ (P : IndObjectPresentation (F.obj i)),
    ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
⊢ ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))

Tactic:
obtain ⟨⟨P⟩⟩ := hF i