-- Mathlib/CategoryTheory/Limits/Indization/FilteredColimits.lean
theorem isFiltered [IsFiltered I] (hF : ∀ i, IsIndObject (F.obj i)) :
    IsFiltered (CostructuredArrow yoneda (colimit F))

Proof state:
case intro.mk'.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
inst✝ : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x✝¹ : SmallCategory J
x✝ : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ : Nonempty (limit (G.op ⋙ (toOver yoneda (colimit F)).op ⋙ yoneda.obj (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙
        yoneda.obj ((colimit.toCostructuredArrow F ⋙ toOver F (colimit F)).obj i)))
P : IndObjectPresentation (F.obj i)
hc : IsColimit ((Over.map (colimit.ι F i)).mapCocone P.cocone.toOver) :=
  isColimitOfPreserves (Over.map (colimit.ι F i)) (Over.isColimitToOver P.coconeIsColimit)
k : P.I
hk :
  Nonempty
    (limit
      (G.op ⋙
        (toOver yoneda (colimit F)).op ⋙
          yoneda.obj
            ((toOver yoneda (colimit F)).obj
              ((pre P.F yoneda (colimit F)).obj
                ((map (colimit.ι F i)).obj (mk (P.cocone.toCostructuredArrow.obj k).hom))))))
htO : (toOver yoneda (colimit F)).FullyFaithful
q : (X : CostructuredArrow yoneda (colimit F)) →
  (toOver yoneda (colimit F)).op ⋙ yoneda.obj ((toOver yoneda (colimit F)).obj X) ≅
    yoneda.obj X ⋙ uliftFunctor.{u, v} :=
  htO.homNatIsoMaxRight
t' :
  limit
    (G.op ⋙
      yoneda.obj
          ((pre P.F yoneda (colimit F)).obj ((map (colimit.ι F i)).obj (mk (P.cocone.toCostructuredArrow.obj k).hom))) ⋙
        uliftFunctor.{u, v})
⊢ ∃ X, Nonempty (limit (G.op ⋙ yoneda.obj X))

Tactic:
exact ⟨_, ⟨((preservesLimitIso uliftFunctor.{u, v} _).inv t').down⟩⟩

Result:
no goals

Proof state:
case intro.mk'.intro.intro
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
inst✝ : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x✝¹ : SmallCategory J
x✝ : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ : Nonempty (limit (G.op ⋙ (toOver yoneda (colimit F)).op ⋙ yoneda.obj (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙
        yoneda.obj ((colimit.toCostructuredArrow F ⋙ toOver F (colimit F)).obj i)))
P : IndObjectPresentation (F.obj i)
hc : IsColimit ((Over.map (colimit.ι F i)).mapCocone P.cocone.toOver) :=
  isColimitOfPreserves (Over.map (colimit.ι F i)) (Over.isColimitToOver P.coconeIsColimit)
k : P.I
hk :
  Nonempty
    (limit
      (G.op ⋙
        (toOver yoneda (colimit F)).op ⋙
          yoneda.obj
            ((toOver yoneda (colimit F)).obj
              ((pre P.F yoneda (colimit F)).obj
                ((map (colimit.ι F i)).obj (mk (P.cocone.toCostructuredArrow.obj k).hom))))))
⊢ ∃ X, Nonempty (limit (G.op ⋙ yoneda.obj X))

Tactic:
have htO : (toOver yoneda (colimit F)).FullyFaithful := .ofFullyFaithful _

Result:
case intro.mk'.intro.intro
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
inst✝ : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x✝¹ : SmallCategory J
x✝ : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ : Nonempty (limit (G.op ⋙ (toOver yoneda (colimit F)).op ⋙ yoneda.obj (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙
        yoneda.obj ((colimit.toCostructuredArrow F ⋙ toOver F (colimit F)).obj i)))
P : IndObjectPresentation (F.obj i)
hc : IsColimit ((Over.map (colimit.ι F i)).mapCocone P.cocone.toOver) :=
  isColimitOfPreserves (Over.map (colimit.ι F i)) (Over.isColimitToOver P.coconeIsColimit)
k : P.I
hk :
  Nonempty
    (limit
      (G.op ⋙
        (toOver yoneda (colimit F)).op ⋙
          yoneda.obj
            ((toOver yoneda (colimit F)).obj
              ((pre P.F yoneda (colimit F)).obj
                ((map (colimit.ι F i)).obj (mk (P.cocone.toCostructuredArrow.obj k).hom))))))
htO : (toOver yoneda (colimit F)).FullyFaithful
⊢ ∃ X, Nonempty (limit (G.op ⋙ yoneda.obj X))

Proof state:
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
inst✝ : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x✝¹ : SmallCategory J
x✝ : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
⊢ ∃ X, Nonempty (limit (G.op ⋙ yoneda.obj X))

Tactic:
have h₁ : Nonempty (limit (G.op ⋙ (toOver _ _).op ⋙ yoneda.obj (Over.mk (𝟙 (colimit F))))) :=
  ⟨Types.Limit.mk _ (fun j => Over.mkIdTerminal.from _) (by simp)⟩

Result:
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
inst✝ : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x✝¹ : SmallCategory J
x✝ : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ : Nonempty (limit (G.op ⋙ (toOver yoneda (colimit F)).op ⋙ yoneda.obj (Over.mk (𝟙 (colimit F)))))
⊢ ∃ X, Nonempty (limit (G.op ⋙ yoneda.obj X))

Proof state:
case intro.mk'.intro.intro
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
inst✝ : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x✝¹ : SmallCategory J
x✝ : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ : Nonempty (limit (G.op ⋙ (toOver yoneda (colimit F)).op ⋙ yoneda.obj (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙
        yoneda.obj ((colimit.toCostructuredArrow F ⋙ toOver F (colimit F)).obj i)))
P : IndObjectPresentation (F.obj i)
hc : IsColimit ((Over.map (colimit.ι F i)).mapCocone P.cocone.toOver) :=
  isColimitOfPreserves (Over.map (colimit.ι F i)) (Over.isColimitToOver P.coconeIsColimit)
k : P.I
hk :
  Nonempty
    (limit
      (G.op ⋙
        (toOver yoneda (colimit F)).op ⋙
          yoneda.obj
            ((toOver yoneda (colimit F)).obj
              ((pre P.F yoneda (colimit F)).obj
                ((map (colimit.ι F i)).obj (mk (P.cocone.toCostructuredArrow.obj k).hom))))))
htO : (toOver yoneda (colimit F)).FullyFaithful
⊢ ∃ X, Nonempty (limit (G.op ⋙ yoneda.obj X))

Tactic:
let q := htO.homNatIsoMaxRight

Result:
case intro.mk'.intro.intro
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
inst✝ : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x✝¹ : SmallCategory J
x✝ : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ : Nonempty (limit (G.op ⋙ (toOver yoneda (colimit F)).op ⋙ yoneda.obj (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙
        yoneda.obj ((colimit.toCostructuredArrow F ⋙ toOver F (colimit F)).obj i)))
P : IndObjectPresentation (F.obj i)
hc : IsColimit ((Over.map (colimit.ι F i)).mapCocone P.cocone.toOver) :=
  isColimitOfPreserves (Over.map (colimit.ι F i)) (Over.isColimitToOver P.coconeIsColimit)
k : P.I
hk :
  Nonempty
    (limit
      (G.op ⋙
        (toOver yoneda (colimit F)).op ⋙
          yoneda.obj
            ((toOver yoneda (colimit F)).obj
              ((pre P.F yoneda (colimit F)).obj
                ((map (colimit.ι F i)).obj (mk (P.cocone.toCostructuredArrow.obj k).hom))))))
htO : (toOver yoneda (colimit F)).FullyFaithful
q : (X : CostructuredArrow yoneda (colimit F)) →
  (toOver yoneda (colimit F)).op ⋙ yoneda.obj ((toOver yoneda (colimit F)).obj X) ≅
    yoneda.obj X ⋙ uliftFunctor.{u, v} :=
  htO.homNatIsoMaxRight
⊢ ∃ X, Nonempty (limit (G.op ⋙ yoneda.obj X))