awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Presentable.IsCardinalFiltered
Goal:
J₁ : Type u
inst : Category.{v, u} J₁
J₂ : Type u'
inst_1 : Category.{v', u'} J₂
F : Functor.{v, v', u, u'} J₁ J₂
inst_2 : F.Final
κ : Cardinal.{w}
inst_3 : Fact κ.IsRegular
inst_4 : IsCardinalFiltered.{w, v, u} J₁ κ
this : IsFiltered.{v, u} J₁
h₁ : ∀ (d : J₂), ∃ (c : J₁), Nonempty.{v' + 1} (d ⟶ F.obj c)
h₂ : ∀ {d : J₂} {c : J₁} (s s' : d ⟶ F.obj c), ∃ (c' : J₁) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t
ι : Type w
j k : J₂
f : ι → (j ⟶ k)
hι : HasCardinalLT ι κ
h : IsEmpty ι
neg : ∀ (j' : J₁) (a : j ⟶ F.obj j'), ∃ (l : J₂) (a : k ⟶ l) (b : j ⟶ l), ∀ (i : ι), f i ≫ a = b
⊢ ∃ (l : J₂) (a : k ⟶ l) (b : j ⟶ l), ∀ (i : ι), f i ≫ a = b

Tactic:
obtain ⟨j', ⟨a⟩⟩ := h₁ j
-- Module: Mathlib.CategoryTheory.Presentable.IsCardinalFiltered
Goal:
J₁ : Type u
inst : Category.{v, u} J₁
J₂ : Type u'
inst_1 : Category.{v', u'} J₂
F : Functor.{v, v', u, u'} J₁ J₂
inst_2 : F.Final
κ : Cardinal.{w}
inst_3 : Fact κ.IsRegular
inst_4 : IsCardinalFiltered.{w, v, u} J₁ κ
this : IsFiltered.{v, u} J₁
h₁ : ∀ (d : J₂), ∃ (c : J₁), Nonempty.{v' + 1} (d ⟶ F.obj c)
h₂ : ∀ {d : J₂} {c : J₁} (s s' : d ⟶ F.obj c), ∃ (c' : J₁) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t
ι : Type w
j k : J₂
f : ι → (j ⟶ k)
hι : HasCardinalLT ι κ
h : IsEmpty ι
j' : J₁
a : j ⟶ F.obj j'
neg : ∀ (k' : J₁) (b : k ⟶ F.obj k'), ∃ (l : J₂) (a : k ⟶ l) (b : j ⟶ l), ∀ (i : ι), f i ≫ a = b
⊢ ∃ (l : J₂) (a : k ⟶ l) (b : j ⟶ l), ∀ (i : ι), f i ≫ a = b

Tactic:
obtain ⟨k', ⟨b⟩⟩ := h₁ k
-- Module: Mathlib.CategoryTheory.Presentable.IsCardinalFiltered
Goal:
J₁ : Type u
inst : Category.{v, u} J₁
J₂ : Type u'
inst_1 : Category.{v', u'} J₂
F : Functor.{v, v', u, u'} J₁ J₂
inst_2 : F.Final
κ : Cardinal.{w}
inst_3 : Fact κ.IsRegular
inst_4 : IsCardinalFiltered.{w, v, u} J₁ κ
this : IsFiltered.{v, u} J₁
h₁ : ∀ (d : J₂), ∃ (c : J₁), Nonempty.{v' + 1} (d ⟶ F.obj c)
h₂ : ∀ {d : J₂} {c : J₁} (s s' : d ⟶ F.obj c), ∃ (c' : J₁) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t
ι : Type w
j k : J₂
f : ι → (j ⟶ k)
hι : HasCardinalLT ι κ
h : IsEmpty ι
j' : J₁
a : j ⟶ F.obj j'
k' : J₁
b : k ⟶ F.obj k'
⊢ ∀ (i : ι),
    Eq (α := j ⟶ F.obj (IsFiltered.max.{v, u} j' k')) (f i ≫ b ≫ F.map (IsFiltered.rightToMax j' k'))
      (a ≫ F.map (IsFiltered.leftToMax j' k'))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Presentable.IsCardinalFiltered
Goal:
J₁ : Type u
inst : Category.{v, u} J₁
J₂ : Type u'
inst_1 : Category.{v', u'} J₂
F : Functor.{v, v', u, u'} J₁ J₂
inst_2 : F.Final
κ : Cardinal.{w}
inst_3 : Fact κ.IsRegular
inst_4 : IsCardinalFiltered.{w, v, u} J₁ κ
this : IsFiltered.{v, u} J₁
h₁ : ∀ (d : J₂), ∃ (c : J₁), Nonempty.{v' + 1} (d ⟶ F.obj c)
h₂ : ∀ {d : J₂} {c : J₁} (s s' : d ⟶ F.obj c), ∃ (c' : J₁) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t
ι : Type w
j k : J₂
f : ι → (j ⟶ k)
hι : HasCardinalLT ι κ
h : IsEmpty ι
j' : J₁
a : j ⟶ F.obj j'
k' : J₁
b : k ⟶ F.obj k'
⊢ ∃ (l : J₂) (a : k ⟶ l) (b : j ⟶ l), ∀ (i : ι), f i ≫ a = b

Tactic:
exact ⟨F.obj (IsFiltered.max j' k'), b ≫ F.map (IsFiltered.rightToMax _ _),
        a ≫ F.map (IsFiltered.leftToMax _ _), by simp⟩
-- Module: Mathlib.CategoryTheory.Presentable.IsCardinalFiltered
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : HasTerminal.{v, u} J
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
⊢ IsCardinalFiltered.{w, v, u} J κ

Tactic:
terminalIsTerminal.isCardinalFiltered _

Premises used: CategoryTheory.Limits.terminalIsTerminal, CategoryTheory.Limits.IsTerminal.isCardinalFiltered
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
x : c.pt.obj cX.pt
⊢ IsFiltered J

Tactic:
have := isFiltered_of_isCardinalFiltered J κ
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
x : c.pt.obj cX.pt
this : IsFiltered J
obtain :
  ∀ (y : (↑(F ⋙ obj (evaluation C (Type w')) cX.pt).sections : Type (max w' u')))
    (hy :
      (Equiv.symm (Types.isLimitEquivSections (hc cX.pt)) :
            (a : (↑(F ⋙ (evaluation C (Type w')).obj cX.pt).sections : Type (max w' u'))) →
              (((evaluation C (Type w')).obj cX.pt).mapCone c).pt)
          y =
        x),
    ∃ (j : J) (x' : c.pt.obj (X.obj j)),
      x =
        (ConcreteCategory.hom (FC := TypeCat.Fun) ((c.pt.mapCocone cX).ι.app j) :
            (a : (fun (X : Type w') ↦ X) ((X ⋙ c.pt).obj j)) →
              (fun (X : Type w') ↦ X) (((const J).obj (c.pt.mapCocone cX).pt).obj j))
          x'
⊢ ∃ (j : J) (x' : c.pt.obj (X.obj j)),
    x =
      (ConcreteCategory.hom (FC := TypeCat.Fun) ((c.pt.mapCocone cX).ι.app j) :
          (a : (fun (X : Type w') ↦ X) ((X ⋙ c.pt).obj j)) →
            (fun (X : Type w') ↦ X) (((const J).obj (c.pt.mapCocone cX).pt).obj j))
        x'

Tactic:
obtain ⟨y, hy⟩ := (Types.isLimitEquivSections (hc cX.pt)).symm.surjective x

Premises used: CategoryTheory.Limits.Types.isLimitEquivSections, Equiv.surjective, Equiv.symm, CategoryTheory.Limits.Cocone.pt
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
x : c.pt.obj cX.pt
this : IsFiltered J
y : (↑(F ⋙ obj (evaluation C (Type w')) cX.pt).sections : Type (max w' u'))
hy :
  (Equiv.symm (Types.isLimitEquivSections (hc cX.pt)) :
        (a : (↑(F ⋙ (evaluation C (Type w')).obj cX.pt).sections : Type (max w' u'))) →
          (((evaluation C (Type w')).obj cX.pt).mapCone c).pt)
      y =
    x
⊢ ∀ (k : K),
    ∃ (j : J) (y_1 : (fun (X : Type w') ↦ X) ((X ⋙ F.obj k).obj j)),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (((F.obj k).mapCocone cX).ι.app j) :
            (a : (fun (X : Type w') ↦ X) ((X ⋙ F.obj k).obj j)) →
              (fun (X : Type w') ↦ X) (((const J).obj ((F.obj k).mapCocone cX).pt).obj j))
          y_1 =
        (↑y : (j : K) → (F ⋙ (evaluation C (Type w')).obj cX.pt).obj j) k

Tactic:
have H (k : K) := Types.jointly_surjective_of_isColimit (hF k) (y.1 k)
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
x : c.pt.obj cX.pt
this : IsFiltered J
y : (↑(F ⋙ obj (evaluation C (Type w')) cX.pt).sections : Type (max w' u'))
hy :
  (Equiv.symm (Types.isLimitEquivSections (hc cX.pt)) :
        (a : (↑(F ⋙ (evaluation C (Type w')).obj cX.pt).sections : Type (max w' u'))) →
          (((evaluation C (Type w')).obj cX.pt).mapCone c).pt)
      y =
    x
obtain :
  ∀ (j₀ : J) (z : (k : K) → (F.obj k).obj (X.obj j₀))
    (hz :
      ∀ (k : K),
        (↑y : (j : K) → (F ⋙ (evaluation C (Type w')).obj cX.pt).obj j) k =
          (ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k).map (cX.ι.app j₀)) :
              (a : (fun (X : Type w') ↦ X) ((F.obj k).obj (X.obj j₀))) →
                (fun (X : Type w') ↦ X) ((F.obj k).obj (((const J).obj cX.pt).obj j₀)))
            (z k)),
    ∃ (j : J) (x' : c.pt.obj (X.obj j)),
      x =
        (ConcreteCategory.hom (FC := TypeCat.Fun) ((c.pt.mapCocone cX).ι.app j) :
            (a : (fun (X : Type w') ↦ X) ((X ⋙ c.pt).obj j)) →
              (fun (X : Type w') ↦ X) (((const J).obj (c.pt.mapCocone cX).pt).obj j))
          x'
⊢ ∃ (j : J) (x' : c.pt.obj (X.obj j)),
    x =
      (ConcreteCategory.hom (FC := TypeCat.Fun) ((c.pt.mapCocone cX).ι.app j) :
          (a : (fun (X : Type w') ↦ X) ((X ⋙ c.pt).obj j)) →
            (fun (X : Type w') ↦ X) (((const J).obj (c.pt.mapCocone cX).pt).obj j))
        x'

Tactic:
obtain ⟨j₀, z, hz⟩ : ∃ (j₀ : J) (z : (k : K) → (F.obj k).obj (X.obj j₀)),
      ∀ (k : K), y.1 k = (F.obj k).map (cX.ι.app j₀) (z k) := by
    have H (k : K) := Types.jointly_surjective_of_isColimit (hF k) (y.1 k)
    let j (k : K) : J := (H k).choose
    let z (k : K) : (F.obj k).obj (X.obj (j k)) := (H k).choose_spec.choose
    have hz (k : K) : (F.obj k).map (cX.ι.app (j k)) (z k) = y.1 k :=
      (H k).choose_spec.choose_spec
    exact ⟨IsCardinalFiltered.max j (hasCardinalLT_of_hasCardinalLT_arrow hK),
      fun k ↦ (F.obj k).map (X.map (IsCardinalFiltered.toMax j _ k)) (z k),
        fun k ↦ by rw [← hz, ← comp_apply, ← Functor.map_comp, cX.w]; rfl⟩
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
x : c.pt.obj cX.pt
this : IsFiltered J
y : (↑(F ⋙ obj (evaluation C (Type w')) cX.pt).sections : Type (max w' u'))
hy :
  (Equiv.symm (Types.isLimitEquivSections (hc cX.pt)) :
        (a : (↑(F ⋙ (evaluation C (Type w')).obj cX.pt).sections : Type (max w' u'))) →
          (((evaluation C (Type w')).obj cX.pt).mapCone c).pt)
      y =
    x
j₀ : J
z : (k : K) → (F.obj k).obj (X.obj j₀)
hz :
  ∀ (k : K),
    (↑y : (j : K) → (F ⋙ (evaluation C (Type w')).obj cX.pt).obj j) k =
      (ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k).map (cX.ι.app j₀)) :
          (a : (fun (X : Type w') ↦ X) ((F.obj k).obj (X.obj j₀))) →
            (fun (X : Type w') ↦ X) ((F.obj k).obj (((const J).obj cX.pt).obj j₀)))
        (z k)
k k' : K
φ : k ⟶ k'
dsimp :
  Eq (α := (F.obj k').obj cX.pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (cX.ι.app j₀)) :
          (a : (F.obj k').obj (X.obj j₀)) → (F.obj k').obj cX.pt)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.map φ).app (X.obj j₀)) :
              (a : (F.obj k).obj (X.obj j₀)) → (F.obj k').obj (X.obj j₀))
            (z k) :
          (F.obj k').obj (X.obj j₀)) :
      (F.obj k').obj cX.pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (cX.ι.app j₀)) :
          (a : (F.obj k').obj (X.obj j₀)) → (F.obj k').obj cX.pt)
        (z k') :
      (F.obj k').obj cX.pt)
⊢ Eq (α := (fun (X : Type w') ↦ X) (obj (obj (const J) ((F.obj k').mapCocone cX).pt) j₀))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F.obj k').mapCocone cX).ι.app j₀) :
          (a : (fun (X : Type w') ↦ X) ((X ⋙ F.obj k').obj j₀)) →
            (fun (X : Type w') ↦ X) (((const J).obj ((F.obj k').mapCocone cX).pt).obj j₀))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.map φ).app (X.obj j₀)) :
              (a : (fun (X : Type w') ↦ X) ((F.obj k).obj (X.obj j₀))) →
                (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀)))
            (z k) :
          (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀))) :
      (fun (X : Type w') ↦ X) (obj (obj (const J) ((F.obj k').mapCocone cX).pt) j₀))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F.obj k').mapCocone cX).ι.app j₀) :
          (a : (fun (X : Type w') ↦ X) ((X ⋙ F.obj k').obj j₀)) →
            (fun (X : Type w') ↦ X) (((const J).obj ((F.obj k').mapCocone cX).pt).obj j₀))
        (z k') :
      (fun (X : Type w') ↦ X) (obj (obj (const J) ((F.obj k').mapCocone cX).pt) j₀))

Tactic:
dsimp at hz ⊢
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
x : c.pt.obj cX.pt
this : IsFiltered J
y : (↑(F ⋙ obj (evaluation C (Type w')) cX.pt).sections : Type (max w' u'))
hy :
  (Equiv.symm (Types.isLimitEquivSections (hc cX.pt)) :
        (a : (↑(F ⋙ (evaluation C (Type w')).obj cX.pt).sections : Type (max w' u'))) →
          (((evaluation C (Type w')).obj cX.pt).mapCone c).pt)
      y =
    x
j₀ : J
z : (k : K) → (F.obj k).obj (X.obj j₀)
hz :
  ∀ (k : K),
    (↑y : (j : K) → (F.obj j).obj cX.pt) k =
      (ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k).map (cX.ι.app j₀)) :
          (a : (F.obj k).obj (X.obj j₀)) → (F.obj k).obj cX.pt)
        (z k)
k k' : K
φ : k ⟶ k'
⊢ Eq (α := (F.obj k').obj cX.pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (cX.ι.app j₀)) :
          (a : (F.obj k').obj (X.obj j₀)) → (F.obj k').obj cX.pt)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.map φ).app (X.obj j₀)) :
              (a : (F.obj k).obj (X.obj j₀)) → (F.obj k').obj (X.obj j₀))
            (z k) :
          (F.obj k').obj (X.obj j₀)) :
      (F.obj k').obj cX.pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (cX.ι.app j₀)) :
          (a : (F.obj k').obj (X.obj j₀)) → (F.obj k').obj cX.pt)
        (z k') :
      (F.obj k').obj cX.pt)

Tactic:
simpa only [← NatTrans.naturality_apply, ← hz] using y.2 φ

Premises used: CategoryTheory.NatTrans.naturality_apply, Subtype.property
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
x : c.pt.obj cX.pt
this : IsFiltered J
y : (↑(F ⋙ obj (evaluation C (Type w')) cX.pt).sections : Type (max w' u'))
hy :
  (Equiv.symm (Types.isLimitEquivSections (hc cX.pt)) :
        (a : (↑(F ⋙ (evaluation C (Type w')).obj cX.pt).sections : Type (max w' u'))) →
          (((evaluation C (Type w')).obj cX.pt).mapCone c).pt)
      y =
    x
j₀ : J
z : (k : K) → (F.obj k).obj (X.obj j₀)
hz :
  ∀ (k : K),
    (↑y : (j : K) → (F ⋙ (evaluation C (Type w')).obj cX.pt).obj j) k =
      (ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k).map (cX.ι.app j₀)) :
          (a : (fun (X : Type w') ↦ X) ((F.obj k).obj (X.obj j₀))) →
            (fun (X : Type w') ↦ X) ((F.obj k).obj (((const J).obj cX.pt).obj j₀)))
        (z k)
⊢ ∀ {k k' : K} (φ : k ⟶ k'),
    ∃ (j : J) (f : j₀ ⟶ j),
      Eq (α := (fun (X : Type w') ↦ X) ((X ⋙ F.obj k').obj j))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((X ⋙ F.obj k').map f) :
              (a : (fun (X : Type w') ↦ X) ((X ⋙ F.obj k').obj j₀)) → (fun (X : Type w') ↦ X) ((X ⋙ F.obj k').obj j))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.map φ).app (X.obj j₀)) :
                  (a : (fun (X : Type w') ↦ X) ((F.obj k).obj (X.obj j₀))) →
                    (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀)))
                (z k) :
              (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀))) :
          (fun (X : Type w') ↦ X) ((X ⋙ F.obj k').obj j))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((X ⋙ F.obj k').map f) :
              (a : (fun (X : Type w') ↦ X) ((X ⋙ F.obj k').obj j₀)) → (fun (X : Type w') ↦ X) ((X ⋙ F.obj k').obj j))
            (z k') :
          (fun (X : Type w') ↦ X) ((X ⋙ F.obj k').obj j))

Tactic:
have H {k k' : K} (φ : k ⟶ k') :=
      (Types.FilteredColimit.isColimit_eq_iff' (ht := hF k')
        (x := (F.map φ).app _ (z k)) (y := z k')).1 (by
          dsimp at hz ⊢
          simpa only [← NatTrans.naturality_apply, ← hz] using y.2 φ)
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
j : J
x₁ x₂ : c.pt.obj (X.obj j)
h :
  Eq (α := (fun (X : Type w') ↦ X) (c.pt.obj (obj.{w, v, w, u} (obj (const J) cX.pt) j)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.pt.map (cX.ι.app j)) :
          (a : (fun (X : Type w') ↦ X) (c.pt.obj (X.obj j))) →
            (fun (X : Type w') ↦ X) (c.pt.obj (((const J).obj cX.pt).obj j)))
        x₁ :
      (fun (X : Type w') ↦ X) (c.pt.obj (obj.{w, v, w, u} (obj (const J) cX.pt) j)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.pt.map (cX.ι.app j)) :
          (a : (fun (X : Type w') ↦ X) (c.pt.obj (X.obj j))) →
            (fun (X : Type w') ↦ X) (c.pt.obj (((const J).obj cX.pt).obj j)))
        x₂ :
      (fun (X : Type w') ↦ X) (c.pt.obj (obj.{w, v, w, u} (obj (const J) cX.pt) j)))
⊢ IsFiltered J

Tactic:
have := isFiltered_of_isCardinalFiltered J κ
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
j : J
x₁ x₂ : c.pt.obj (X.obj j)
h :
  Eq (α := (fun (X : Type w') ↦ X) (c.pt.obj (obj.{w, v, w, u} (obj (const J) cX.pt) j)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.pt.map (cX.ι.app j)) :
          (a : (fun (X : Type w') ↦ X) (c.pt.obj (X.obj j))) →
            (fun (X : Type w') ↦ X) (c.pt.obj (((const J).obj cX.pt).obj j)))
        x₁ :
      (fun (X : Type w') ↦ X) (c.pt.obj (obj.{w, v, w, u} (obj (const J) cX.pt) j)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.pt.map (cX.ι.app j)) :
          (a : (fun (X : Type w') ↦ X) (c.pt.obj (X.obj j))) →
            (fun (X : Type w') ↦ X) (c.pt.obj (((const J).obj cX.pt).obj j)))
        x₂ :
      (fun (X : Type w') ↦ X) (c.pt.obj (obj.{w, v, w, u} (obj (const J) cX.pt) j)))
this : IsFiltered J
y₁ : (↑(F ⋙ obj (evaluation C (Type w')) (X.obj j)).sections : Type (max w' u')) :=
  (Types.isLimitEquivSections (hc (X.obj j)) :
      (a : (((evaluation C (Type w')).obj (X.obj j)).mapCone c).pt) →
        (↑(F ⋙ (evaluation C (Type w')).obj (X.obj j)).sections : Type (max w' u')))
    x₁
y₂ : (↑(F ⋙ obj (evaluation C (Type w')) (X.obj j)).sections : Type (max w' u')) :=
  (Types.isLimitEquivSections (hc (X.obj j)) :
      (a : (((evaluation C (Type w')).obj (X.obj j)).mapCone c).pt) →
        (↑(F ⋙ (evaluation C (Type w')).obj (X.obj j)).sections : Type (max w' u')))
    x₂
⊢ (Equiv.symm (Types.isLimitEquivSections (hc (X.obj j))) :
        (a : (↑(F ⋙ (evaluation C (Type w')).obj (X.obj j)).sections : Type (max w' u'))) →
          (((evaluation C (Type w')).obj (X.obj j)).mapCone c).pt)
      y₁ =
    x₁

Tactic:
simp [y₁]
-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
j : J
x₁ x₂ : c.pt.obj (X.obj j)
h :
  Eq (α := (fun (X : Type w') ↦ X) (c.pt.obj (obj.{w, v, w, u} (obj (const J) cX.pt) j)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.pt.map (cX.ι.app j)) :
          (a : (fun (X : Type w') ↦ X) (c.pt.obj (X.obj j))) →
            (fun (X : Type w') ↦ X) (c.pt.obj (((const J).obj cX.pt).obj j)))
        x₁ :
      (fun (X : Type w') ↦ X) (c.pt.obj (obj.{w, v, w, u} (obj (const J) cX.pt) j)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.pt.map (cX.ι.app j)) :
          (a : (fun (X : Type w') ↦ X) (c.pt.obj (X.obj j))) →
            (fun (X : Type w') ↦ X) (c.pt.obj (((const J).obj cX.pt).obj j)))
        x₂ :
      (fun (X : Type w') ↦ X) (c.pt.obj (obj.{w, v, w, u} (obj (const J) cX.pt) j)))
this : IsFiltered J
y₁ : (↑(F ⋙ obj (evaluation C (Type w')) (X.obj j)).sections : Type (max w' u')) :=
  (Types.isLimitEquivSections (hc (X.obj j)) :
      (a : (((evaluation C (Type w')).obj (X.obj j)).mapCone c).pt) →
        (↑(F ⋙ (evaluation C (Type w')).obj (X.obj j)).sections : Type (max w' u')))
    x₁
y₂ : (↑(F ⋙ obj (evaluation C (Type w')) (X.obj j)).sections : Type (max w' u')) :=
  (Types.isLimitEquivSections (hc (X.obj j)) :
      (a : (((evaluation C (Type w')).obj (X.obj j)).mapCone c).pt) →
        (↑(F ⋙ (evaluation C (Type w')).obj (X.obj j)).sections : Type (max w' u')))
    x₂
⊢ (Equiv.symm (Types.isLimitEquivSections (hc (X.obj j))) :
        (a : (↑(F ⋙ (evaluation C (Type w')).obj (X.obj j)).sections : Type (max w' u'))) →
          (((evaluation C (Type w')).obj (X.obj j)).mapCone c).pt)
      y₁ =
    x₁

Tactic:
have hy₁ : (Types.isLimitEquivSections (hc (X.obj j))).symm y₁ = x₁ := by simp [y₁]