-- Mathlib/CategoryTheory/Limits/Indization/FilteredColimits.lean
theorem isIndObject_colimit (I : Type v) [SmallCategory I] [IsFiltered I]
    (F : I ⥤ Cᵒᵖ ⥤ Type v) (hF : ∀ i, IsIndObject (F.obj i)) : IsIndObject (colimit F)

Proof state:
case intro.intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy : yonedaEquiv A.hom = (colimit.ι F i).app (Opposite.op A.left) y
y' : CostructuredArrow yoneda (F.obj i) := mk (yonedaEquiv.symm y)
x : y' ⟶ j i y'
⊢ A ⟶ (map (colimit.ι F i)).obj (j i y')

Tactic:
refine ?_ ≫ (map (colimit.ι F i)).map x

Result:
case intro.intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy : yonedaEquiv A.hom = (colimit.ι F i).app (Opposite.op A.left) y
y' : CostructuredArrow yoneda (F.obj i) := mk (yonedaEquiv.symm y)
x : y' ⟶ j i y'
⊢ A ⟶ (map (colimit.ι F i)).obj y'

Proof state:
case intro.intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy : yonedaEquiv A.hom = (colimit.ι F i).app (Opposite.op A.left) y
y' : CostructuredArrow yoneda (F.obj i) := mk (yonedaEquiv.symm y)
x : y' ⟶ j i y'
⊢ yonedaEquiv (yoneda.map (𝟙 A.left) ≫ ((map (colimit.ι F i)).obj y').hom) = yonedaEquiv A.hom

Tactic:
simp [-EmbeddingLike.apply_eq_iff_eq, hy, yonedaEquiv_comp, y']

Result:
no goals

Proof state:
case intro.intro
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy : yonedaEquiv A.hom = (colimit.ι F i).app (Opposite.op A.left) y
⊢ ∃ j ∈ ⋃ i, (map (colimit.ι F i)).obj '' s i, Nonempty (A ⟶ j)

Tactic:
let y' : CostructuredArrow yoneda (F.obj i) := mk (yonedaEquiv.symm y)

Result:
case intro.intro
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy : yonedaEquiv A.hom = (colimit.ι F i).app (Opposite.op A.left) y
y' : CostructuredArrow yoneda (F.obj i) := mk (yonedaEquiv.symm y)
⊢ ∃ j ∈ ⋃ i, (map (colimit.ι F i)).obj '' s i, Nonempty (A ⟶ j)

Proof state:
case intro.intro.intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy : yonedaEquiv A.hom = (colimit.ι F i).app (Opposite.op A.left) y
y' : CostructuredArrow yoneda (F.obj i) := mk (yonedaEquiv.symm y)
x : y' ⟶ j i y'
⊢ ∃ i_1, ∃ x ∈ s i_1, (map (colimit.ι F i_1)).obj x = (map (colimit.ι F i)).obj (j i y')

Tactic:
exact ⟨i, j i y', hjs _ _, rfl⟩

Result:
no goals

Proof state:
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
⊢ IsIndObject (colimit F)

Tactic:
refine (isIndObject_iff _).mpr ⟨this, ?_⟩

Result:
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
⊢ FinallySmall (CostructuredArrow yoneda (colimit F))

Proof state:
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
⊢ FinallySmall (CostructuredArrow yoneda (colimit F))

Tactic:
refine finallySmall_of_small_weakly_terminal_set
  (⋃ i, (map (colimit.ι F i)).obj '' (s i)) (fun A => ?_)

Result:
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
⊢ ∃ j ∈ ⋃ i, (map (colimit.ι F i)).obj '' s i, Nonempty (A ⟶ j)