awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
Goal:
C : Type u
inst : Category.{v, u} C
ι : Type u_3
ι' : Type u_4
S : C
X : ι → C
a : Cofan.{u_3, v, u} X
hau : IsUniversalColimit a
Y : ι' → C
b : Cofan.{u_4, v, u} Y
hbu : IsUniversalColimit b
f : (i : ι) → X i ⟶ S
g : (i : ι') → Y i ⟶ S
u : a.pt ⟶ S
v : b.pt ⟶ S
inst_1 : ∀ (i : ι), HasPullback (f i) v
inst_2 : HasPullback u v
P : ι × ι' → C
q₁ : (i : ι) → (j : ι') → P (i, j) ⟶ X i
q₂ : (i : ι) → (j : ι') → P (i, j) ⟶ Y j
hP : ∀ (i : ι) (j : ι'), IsPullback (q₁ i j) (q₂ i j) (f i) (g j)
d : Cofan P
hd : IsColimit d
hu : ∀ (i : ι), a.inj i ≫ u = f i
hv : ∀ (i : ι'), b.inj i ≫ v = g i
c : Cofan P :=
  Cofan.mk (pullback u v) fun (p : ι × ι') ↦ pullback.lift (q₁ p.1 p.2 ≫ a.inj p.1) (q₂ p.1 p.2 ≫ b.inj p.2) ⋯
hc : IsColimit c
⊢ pullback.snd u v ≫ Iso.hom (Iso.refl b.pt) =
    (hc.coconePointUniqueUpToIso hd).hom ≫ Cofan.IsColimit.desc hd fun (p : ι × ι') ↦ q₂ p.1 p.2 ≫ b.inj p.2

Tactic:
Cofan.IsColimit.hom_ext hc _ _ fun i ↦ ?_

Premises used: CategoryTheory.Limits.Cofan.IsColimit.hom_ext, CategoryTheory.Limits.limit
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
Goal:
C : Type u
inst : Category.{v, u} C
ι : Type u_3
ι' : Type u_4
S : C
X : ι → C
a : Cofan.{u_3, v, u} X
hau : IsUniversalColimit a
Y : ι' → C
b : Cofan.{u_4, v, u} Y
hbu : IsUniversalColimit b
f : (i : ι) → X i ⟶ S
g : (i : ι') → Y i ⟶ S
u : a.pt ⟶ S
v : b.pt ⟶ S
inst_1 : ∀ (i : ι), HasPullback (f i) v
inst_2 : HasPullback u v
P : ι × ι' → C
q₁ : (i : ι) → (j : ι') → P (i, j) ⟶ X i
q₂ : (i : ι) → (j : ι') → P (i, j) ⟶ Y j
hP : ∀ (i : ι) (j : ι'), IsPullback (q₁ i j) (q₂ i j) (f i) (g j)
d : Cofan P
hd : IsColimit d
hu : ∀ (i : ι), a.inj i ≫ u = f i
hv : ∀ (i : ι'), b.inj i ≫ v = g i
c : Cofan P :=
  Cofan.mk (pullback u v) fun (p : ι × ι') ↦ pullback.lift (q₁ p.1 p.2 ≫ a.inj p.1) (q₂ p.1 p.2 ≫ b.inj p.2) ⋯
hc : IsColimit c
refine_3 :
  ∀ (i : ι × ι'),
    c.inj i ≫ pullback.snd u v ≫ Iso.hom (Iso.refl b.pt) =
      c.inj i ≫ (hc.coconePointUniqueUpToIso hd).hom ≫ Cofan.IsColimit.desc hd fun (p : ι × ι') ↦ q₂ p.1 p.2 ≫ b.inj p.2
⊢ pullback.snd u v ≫ Iso.hom (Iso.refl b.pt) =
    (hc.coconePointUniqueUpToIso hd).hom ≫ Cofan.IsColimit.desc hd fun (p : ι × ι') ↦ q₂ p.1 p.2 ≫ b.inj p.2

Tactic:
refine Cofan.IsColimit.hom_ext hc _ _ fun i ↦ ?_

Premises used: CategoryTheory.Limits.Cofan.IsColimit.hom_ext
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
Goal:
C : Type u
inst : Category.{v, u} C
ι : Type u_3
ι' : Type u_4
S : C
X : ι → C
a : Cofan.{u_3, v, u} X
hau : IsUniversalColimit a
Y : ι' → C
b : Cofan.{u_4, v, u} Y
hbu : IsUniversalColimit b
f : (i : ι) → X i ⟶ S
g : (i : ι') → Y i ⟶ S
u : a.pt ⟶ S
v : b.pt ⟶ S
inst_1 : ∀ (i : ι), HasPullback (f i) v
inst_2 : HasPullback u v
P : ι × ι' → C
q₁ : (i : ι) → (j : ι') → P (i, j) ⟶ X i
q₂ : (i : ι) → (j : ι') → P (i, j) ⟶ Y j
hP : ∀ (i : ι) (j : ι'), IsPullback (q₁ i j) (q₂ i j) (f i) (g j)
d : Cofan P
hd : IsColimit d
hu : ∀ (i : ι), a.inj i ≫ u = f i
hv : ∀ (i : ι'), b.inj i ≫ v = g i
c : Cofan P :=
  Cofan.mk (pullback u v) fun (p : ι × ι') ↦ pullback.lift (q₁ p.1 p.2 ≫ a.inj p.1) (q₂ p.1 p.2 ≫ b.inj p.2) ⋯
hc : IsColimit c
i : ι × ι'
⊢ c.inj i ≫ pullback.snd u v ≫ Iso.hom (Iso.refl b.pt) =
    c.inj i ≫ (hc.coconePointUniqueUpToIso hd).hom ≫ Cofan.IsColimit.desc hd fun (p : ι × ι') ↦ q₂ p.1 p.2 ≫ b.inj p.2

Tactic:
simpa [Cofan.inj, Cofan.IsColimit.desc] using pullback.lift_snd _ _ _

Premises used: CategoryTheory.Limits.Cofan.IsColimit.desc, CategoryTheory.Limits.pullback.lift_snd, CategoryTheory.Limits.Cofan.inj
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
X : Cᵒᵖ
s : Cocone.{v, v, u, v + 1} (Functor.obj coyoneda X)
Y : C
h :
  ∀ (f : Functor.obj (Functor.obj coyoneda X) Y),
    TypeCat.Fun.toFun
        (hom (FC := TypeCat.Fun)
          (CategoryStruct.comp (Y := Functor.obj (Functor.obj (Functor.const C) (Cocone.pt (colimitCocone X))) Y)
            (NatTrans.app (Cocone.ι (colimitCocone X)) Y)
            (↾fun (x : Cocone.pt (colimitCocone X)) ↦
              (hom (FC := TypeCat.Fun) (s.ι.app (unop X)) :
                  (a : (fun (X : Type v) ↦ X) ((coyoneda.obj X).obj (unop X))) →
                    (fun (X : Type v) ↦ X) (((Functor.const C).obj s.pt).obj (unop X)))
                (𝟙 (unop X)))))
        f =
      (hom (FC := TypeCat.Fun) (s.ι.app Y)).toFun f
⊢ CategoryStruct.comp (Y := Functor.obj (Functor.obj (Functor.const C) (Cocone.pt (colimitCocone X))) Y)
      (NatTrans.app (Cocone.ι (colimitCocone X)) Y)
      (↾fun (x : Cocone.pt (colimitCocone X)) ↦
        (hom (FC := TypeCat.Fun) (s.ι.app (unop X)) :
            (a : (fun (X : Type v) ↦ X) ((coyoneda.obj X).obj (unop X))) →
              (fun (X : Type v) ↦ X) (((Functor.const C).obj s.pt).obj (unop X)))
          (𝟙 (unop X))) =
    s.ι.app Y

Tactic:
ext f
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
X : Cᵒᵖ
s : Cocone.{v, v, u, v + 1} (Functor.obj coyoneda X)
Y : C
f : Functor.obj (Functor.obj coyoneda X) Y
⊢ TypeCat.Fun.toFun
      (hom (FC := TypeCat.Fun)
        (CategoryStruct.comp (Y := Functor.obj (Functor.obj (Functor.const C) (Cocone.pt (colimitCocone X))) Y)
          (NatTrans.app (Cocone.ι (colimitCocone X)) Y)
          (↾fun (x : Cocone.pt (colimitCocone X)) ↦
            (hom (FC := TypeCat.Fun) (s.ι.app (unop X)) :
                (a : (fun (X : Type v) ↦ X) ((coyoneda.obj X).obj (unop X))) →
                  (fun (X : Type v) ↦ X) (((Functor.const C).obj s.pt).obj (unop X)))
              (𝟙 (unop X)))))
      f =
    (hom (FC := TypeCat.Fun) (s.ι.app Y)).toFun f

Tactic:
simpa using congr_hom (s.w f).symm (𝟙 (unop X))

Premises used: CategoryTheory.Limits.Cocone.w, CategoryTheory.ConcreteCategory.congr_hom, Eq.symm
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
X : Cᵒᵖ
s : Cocone.{v, v, u, v + 1} (Functor.obj coyoneda X)
m : Cocone.pt (colimitCocone X) ⟶ s.pt
w : ∀ (j : C), NatTrans.app (Cocone.ι (colimitCocone X)) j ≫ m = s.ι.app j
h :
  (hom (FC := TypeCat.Fun) m).toFun PUnit.unit =
    TypeCat.Fun.toFun
      (hom (FC := TypeCat.Fun)
        (↾fun (x : Cocone.pt (colimitCocone X)) ↦
          (hom (FC := TypeCat.Fun) (s.ι.app (unop X)) :
              (a : (fun (X : Type v) ↦ X) ((coyoneda.obj X).obj (unop X))) →
                (fun (X : Type v) ↦ X) (((Functor.const C).obj s.pt).obj (unop X)))
            (𝟙 (unop X))))
      PUnit.unit
⊢ m =
    ↾fun (x : Cocone.pt (colimitCocone X)) ↦
      (hom (FC := TypeCat.Fun) (s.ι.app (unop X)) :
          (a : (fun (X : Type v) ↦ X) ((coyoneda.obj X).obj (unop X))) →
            (fun (X : Type v) ↦ X) (((Functor.const C).obj s.pt).obj (unop X)))
        (𝟙 (unop X))

Tactic:
ext ⟨⟩
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
X : Cᵒᵖ
s : Cocone.{v, v, u, v + 1} (Functor.obj coyoneda X)
m : Cocone.pt (colimitCocone X) ⟶ s.pt
w : ∀ (j : C), NatTrans.app (Cocone.ι (colimitCocone X)) j ≫ m = s.ι.app j
⊢ (hom (FC := TypeCat.Fun) m).toFun PUnit.unit =
    TypeCat.Fun.toFun
      (hom (FC := TypeCat.Fun)
        (↾fun (x : Cocone.pt (colimitCocone X)) ↦
          (hom (FC := TypeCat.Fun) (s.ι.app (unop X)) :
              (a : (fun (X : Type v) ↦ X) ((coyoneda.obj X).obj (unop X))) →
                (fun (X : Type v) ↦ X) (((Functor.const C).obj s.pt).obj (unop X)))
            (𝟙 (unop X))))
      PUnit.unit

Tactic:
simp [← w]
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
X : Cᵒᵖ
⊢ HasColimit.{v, u, v, v + 1} (Functor.obj coyoneda X)

Tactic:
HasColimit.mk
    { cocone := _
      isColimit := colimitCoconeIsColimit X }

Premises used: CategoryTheory.Limits.HasColimit.mk, CategoryTheory.Limits.ColimitCocone.mk, CategoryTheory.Coyoneda.colimitCocone, CategoryTheory.Coyoneda.colimitCoconeIsColimit
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
X : Cᵒᵖ
⊢ colimit (Functor.obj coyoneda X) ≅ PUnit.{v + 1}

Tactic:
apply colimit.isoColimitCocone
    { cocone := _
      isColimit := colimitCoconeIsColimit X }

Premises used: CategoryTheory.Limits.colimit.isoColimitCocone, CategoryTheory.Limits.ColimitCocone.mk, CategoryTheory.Coyoneda.colimitCocone, CategoryTheory.Coyoneda.colimitCoconeIsColimit
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
X : C
s : (↑(F ⋙ Functor.obj yoneda X).sections : Type (max v w))
x x_1 : J
f : x ⟶ x_1
⊢ Eq (α := Functor.obj.{t, v, w, u} (Functor.obj (Functor.const J) (op X)) x ⟶ F.obj x_1)
    (Functor.map (Functor.obj (Functor.const J) (op X)) f ≫
      Quiver.Hom.op ((↑s : (j : J) → (F ⋙ yoneda.obj X).obj j) x_1))
    (Quiver.Hom.op ((↑s : (j : J) → (F ⋙ yoneda.obj X).obj j) x) ≫ F.map f)

Tactic:
simp [(s.property f).symm]

Premises used: Eq.symm, Subtype.property
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
X : C
c : Cone F
hc : IsLimit c
rw :
  ∀ s ∈ (F ⋙ Functor.obj yoneda X).sections,
    ∃! x : Cone.pt (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c),
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (NatTrans.app (Cone.π (Functor.mapCone (Functor.obj yoneda X) c)) j) :
              (a : (fun (X : Type v) ↦ X) (((Functor.const J).obj ((yoneda.obj X).mapCone c).pt).obj j)) →
                (fun (X : Type v) ↦ X) ((F ⋙ yoneda.obj X).obj j))
            x =
          s j
⊢ Nonempty (IsLimit (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c))

Tactic:
rw [Types.isLimit_iff]

Premises used: CategoryTheory.Limits.Types.isLimit_iff
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
X : C
c : Cone F
hc : IsLimit c
s : (j : J) → (F ⋙ Functor.obj yoneda X).obj j
hs : s ∈ (F ⋙ Functor.obj yoneda X).sections
⊢ ∃! x : Cone.pt (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c),
    ∀ (j : J),
      (hom (FC := TypeCat.Fun) (NatTrans.app (Cone.π (Functor.mapCone (Functor.obj yoneda X) c)) j) :
            (a : (fun (X : Type v) ↦ X) (((Functor.const J).obj ((yoneda.obj X).mapCone c).pt).obj j)) →
              (fun (X : Type v) ↦ X) ((F ⋙ yoneda.obj X).obj j))
          x =
        s j

Tactic:
⟨(hc.lift (Limits.coneOfSectionCompYoneda F X ⟨s, hs⟩)).unop,
      fun j => Quiver.Hom.op_inj (hc.fac (Limits.coneOfSectionCompYoneda F X ⟨s, hs⟩) j),
      fun m hm => Quiver.Hom.op_inj
        (hc.uniq (Limits.coneOfSectionCompYoneda F X ⟨s, hs⟩) _
          (fun j => Quiver.Hom.unop_inj (hm j)))⟩

Premises used: Quiver.Hom.op, Quiver.Hom.op_inj, Quiver.Hom.unop, CategoryTheory.Limits.IsLimit.uniq, Subtype.mk, Quiver.Hom.unop_inj, ExistsUnique, And.intro, Exists.intro, CategoryTheory.Limits.IsLimit.lift, CategoryTheory.Limits.coneOfSectionCompYoneda, CategoryTheory.Limits.IsLimit.fac
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
X : C
c : Cone F
hc : IsLimit c
s : (j : J) → (F ⋙ Functor.obj yoneda X).obj j
hs : s ∈ (F ⋙ Functor.obj yoneda X).sections
⊢ ∃! x : Cone.pt (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c),
    ∀ (j : J),
      (hom (FC := TypeCat.Fun) (NatTrans.app (Cone.π (Functor.mapCone (Functor.obj yoneda X) c)) j) :
            (a : (fun (X : Type v) ↦ X) (((Functor.const J).obj ((yoneda.obj X).mapCone c).pt).obj j)) →
              (fun (X : Type v) ↦ X) ((F ⋙ yoneda.obj X).obj j))
          x =
        s j

Tactic:
exact ⟨(hc.lift (Limits.coneOfSectionCompYoneda F X ⟨s, hs⟩)).unop,
      fun j => Quiver.Hom.op_inj (hc.fac (Limits.coneOfSectionCompYoneda F X ⟨s, hs⟩) j),
      fun m hm => Quiver.Hom.op_inj
        (hc.uniq (Limits.coneOfSectionCompYoneda F X ⟨s, hs⟩) _
          (fun j => Quiver.Hom.unop_inj (hm j)))⟩
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
c : Cone F
hc : (X : C) → IsLimit (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c)
s : Cone F
j : J
⊢ Quiver.Hom.unop
      (Quiver.Hom.op
          ((hom (FC := TypeCat.Fun)
                (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) :
              (a : (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone s).pt) →
                (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone c).pt)
            (𝟙 (unop s.pt))) ≫
        c.π.app j) =
    (s.π.app j).unop

Tactic:
simpa using congr_hom ((hc s.pt.unop).fac ((yoneda.obj s.pt.unop).mapCone s) j) (𝟙 (unop s.pt))

Premises used: CategoryTheory.yoneda, CategoryTheory.Limits.Cone.pt, CategoryTheory.ConcreteCategory.congr_hom, CategoryTheory.Functor.mapCone, Opposite.unop, CategoryTheory.Functor.obj, CategoryTheory.Limits.IsLimit.fac
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
c : Cone F
hc : (X : C) → IsLimit (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c)
s : Cone F
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
a :
  Eq (α := (↑(F ⋙ Functor.obj yoneda (unop s.pt)).sections : Type (max v w)))
    ((Types.isLimitEquivSections (hc (unop s.pt)) :
          (a : ((yoneda.obj (unop s.pt)).mapCone c).pt) → (↑(F ⋙ yoneda.obj (unop s.pt)).sections : Type (max v w)))
        m.unop :
      (↑(F ⋙ Functor.obj yoneda (unop s.pt)).sections : Type (max v w)))
    ((Types.isLimitEquivSections (hc (unop s.pt)) :
          (a : ((yoneda.obj (unop s.pt)).mapCone c).pt) → (↑(F ⋙ yoneda.obj (unop s.pt)).sections : Type (max v w)))
        (Quiver.Hom.unop
          (Quiver.Hom.op
            ((hom (FC := TypeCat.Fun)
                  (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) :
                (a : (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone s).pt) →
                  (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone c).pt)
              (𝟙 (unop s.pt))))) :
      (↑(F ⋙ Functor.obj yoneda (unop s.pt)).sections : Type (max v w)))
⊢ m.unop =
    Quiver.Hom.unop
      (Quiver.Hom.op
        ((hom (FC := TypeCat.Fun) (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) :
            (a : (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone s).pt) →
              (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone c).pt)
          (𝟙 (unop s.pt))))

Tactic:
apply (Types.isLimitEquivSections (hc s.pt.unop)).injective

Premises used: CategoryTheory.Limits.Types.isLimitEquivSections, CategoryTheory.Limits.Cone.pt, Equiv.injective, Opposite.unop
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
c : Cone F
hc : (X : C) → IsLimit (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c)
s : Cone F
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
a :
  ∀ (j : J),
    Eq.{v + 1} (α := (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
      ((↑((Types.isLimitEquivSections (hc (unop s.pt)) :
                (a : ((yoneda.obj (unop s.pt)).mapCone c).pt) →
                  (↑(F ⋙ yoneda.obj (unop s.pt)).sections : Type (max v w)))
              m.unop) :
          (j : J) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
        j)
      ((↑((Types.isLimitEquivSections (hc (unop s.pt)) :
                (a : ((yoneda.obj (unop s.pt)).mapCone c).pt) →
                  (↑(F ⋙ yoneda.obj (unop s.pt)).sections : Type (max v w)))
              (Quiver.Hom.unop
                (Quiver.Hom.op
                  ((hom (FC := TypeCat.Fun)
                        (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) :
                      (a : (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone s).pt) →
                        (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone c).pt)
                    (𝟙 (unop s.pt)))))) :
          (j : J) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
        j)
⊢ Eq (α := (↑(F ⋙ Functor.obj yoneda (unop s.pt)).sections : Type (max v w)))
    ((Types.isLimitEquivSections (hc (unop s.pt)) :
          (a : ((yoneda.obj (unop s.pt)).mapCone c).pt) → (↑(F ⋙ yoneda.obj (unop s.pt)).sections : Type (max v w)))
        m.unop :
      (↑(F ⋙ Functor.obj yoneda (unop s.pt)).sections : Type (max v w)))
    ((Types.isLimitEquivSections (hc (unop s.pt)) :
          (a : ((yoneda.obj (unop s.pt)).mapCone c).pt) → (↑(F ⋙ yoneda.obj (unop s.pt)).sections : Type (max v w)))
        (Quiver.Hom.unop
          (Quiver.Hom.op
            ((hom (FC := TypeCat.Fun)
                  (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) :
                (a : (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone s).pt) →
                  (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone c).pt)
              (𝟙 (unop s.pt))))) :
      (↑(F ⋙ Functor.obj yoneda (unop s.pt)).sections : Type (max v w)))

Tactic:
ext j
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
c : Cone F
hc : (X : C) → IsLimit (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c)
s : Cone F
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
j : J
⊢ Eq (α := (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
    ((hom (FC := TypeCat.Fun)
            (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s) ≫
              NatTrans.app (Cone.π (Functor.mapCone (Functor.obj yoneda (unop s.pt)) c)) j) :
          (a : ((yoneda.obj (unop s.pt)).mapCone s).pt) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
        (𝟙 (unop s.pt)) :
      (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
    (((@hom _ types TypeCat.Fun _ _ _ _ _
              (NatTrans.app (Cone.π (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) j) :
            TypeCat.Fun
              (Cone.pt (F := F ⋙ Functor.obj yoneda (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s))
              ((F ⋙ Functor.obj yoneda (unop s.pt)).obj j)) :
          (a : ((yoneda.obj (unop s.pt)).mapCone s).pt) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
        (𝟙 (unop s.pt)) :
      (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)

Tactic:
have eq := congr_hom ((hc s.pt.unop).fac ((yoneda.obj s.pt.unop).mapCone s) j) (𝟙 (unop s.pt))
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
c : Cone F
hc : (X : C) → IsLimit (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c)
s : Cone F
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
j : J
eq :
  Eq (α := (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
    ((hom (FC := TypeCat.Fun)
            (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s) ≫
              NatTrans.app (Cone.π (Functor.mapCone (Functor.obj yoneda (unop s.pt)) c)) j) :
          (a : ((yoneda.obj (unop s.pt)).mapCone s).pt) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
        (𝟙 (unop s.pt)) :
      (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
    (((@hom _ types TypeCat.Fun _ _ _ _ _
              (NatTrans.app (Cone.π (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) j) :
            TypeCat.Fun
              (Cone.pt (F := F ⋙ Functor.obj yoneda (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s))
              ((F ⋙ Functor.obj yoneda (unop s.pt)).obj j)) :
          (a : ((yoneda.obj (unop s.pt)).mapCone s).pt) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
        (𝟙 (unop s.pt)) :
      (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
a :
  (c.π.app j).unop ≫ m.unop =
    (c.π.app j).unop ≫
      (@hom _ types TypeCat.Fun _ _ _ _ _
            (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) :
          (a : unop s.pt ⟶ unop s.pt) → unop c.pt ⟶ unop s.pt)
        (𝟙 (unop s.pt))
⊢ Eq.{v + 1} (α := (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
    ((↑((Types.isLimitEquivSections (hc (unop s.pt)) :
              (a : ((yoneda.obj (unop s.pt)).mapCone c).pt) → (↑(F ⋙ yoneda.obj (unop s.pt)).sections : Type (max v w)))
            m.unop) :
        (j : J) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
      j)
    ((↑((Types.isLimitEquivSections (hc (unop s.pt)) :
              (a : ((yoneda.obj (unop s.pt)).mapCone c).pt) → (↑(F ⋙ yoneda.obj (unop s.pt)).sections : Type (max v w)))
            (Quiver.Hom.unop
              (Quiver.Hom.op
                ((hom (FC := TypeCat.Fun)
                      (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) :
                    (a : (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone s).pt) →
                      (fun (X : Type v) ↦ X) ((yoneda.obj (unop s.pt)).mapCone c).pt)
                  (𝟙 (unop s.pt)))))) :
        (j : J) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
      j)

Tactic:
dsimp [Types.isLimitEquivSections, Types.sectionOfCone]

Premises used: CategoryTheory.Limits.Types.isLimitEquivSections, CategoryTheory.Limits.Types.sectionOfCone
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J Cᵒᵖ
c : Cone F
hc : (X : C) → IsLimit (F := F ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c)
s : Cone F
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
j : J
eq :
  Eq (α := (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
    ((hom (FC := TypeCat.Fun)
            (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s) ≫
              NatTrans.app (Cone.π (Functor.mapCone (Functor.obj yoneda (unop s.pt)) c)) j) :
          (a : ((yoneda.obj (unop s.pt)).mapCone s).pt) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
        (𝟙 (unop s.pt)) :
      (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
    (((@hom _ types TypeCat.Fun _ _ _ _ _
              (NatTrans.app (Cone.π (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) j) :
            TypeCat.Fun
              (Cone.pt (F := F ⋙ Functor.obj yoneda (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s))
              ((F ⋙ Functor.obj yoneda (unop s.pt)).obj j)) :
          (a : ((yoneda.obj (unop s.pt)).mapCone s).pt) → (F ⋙ yoneda.obj (unop s.pt)).obj j)
        (𝟙 (unop s.pt)) :
      (F ⋙ Functor.obj yoneda (unop s.pt)).obj j)
⊢ (c.π.app j).unop ≫ m.unop =
    (c.π.app j).unop ≫
      (@hom _ types TypeCat.Fun _ _ _ _ _
            (IsLimit.lift (hc (unop s.pt)) (Functor.mapCone (Functor.obj yoneda (unop s.pt)) s)) :
          (a : unop s.pt ⟶ unop s.pt) → unop c.pt ⟶ unop s.pt)
        (𝟙 (unop s.pt))

Tactic:
simp_all [← hm]
-- Module: Mathlib.CategoryTheory.Limits.Yoneda
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
inst_1 : Category.{t, w} J
F : Functor.{t, v, w, u} J C
c : Cocone F
x : (X : C) → IsLimit (F := F.op ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c.op)
h :
  ∀ (x_1 : C),
    (fun (h : IsColimit c) (x : C) ↦
          (isLimitOfPreserves (Functor.obj yoneda x) h.op :
            IsLimit (F := F.op ⋙ Functor.obj yoneda x) (Functor.mapCone (Functor.obj yoneda x) c.op)))
        ((fun (h : (X : C) → IsLimit (F := F.op ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c.op)) ↦
            (yonedaJointlyReflectsLimits F.op c.op h).unop)
          x)
        x_1 =
      x x_1
⊢ (fun (h : IsColimit c) (x : C) ↦
        (isLimitOfPreserves (Functor.obj yoneda x) h.op :
          IsLimit (F := F.op ⋙ Functor.obj yoneda x) (Functor.mapCone (Functor.obj yoneda x) c.op)))
      ((fun (h : (X : C) → IsLimit (F := F.op ⋙ Functor.obj yoneda X) (Functor.mapCone (Functor.obj yoneda X) c.op)) ↦
          (yonedaJointlyReflectsLimits F.op c.op h).unop)
        x) =
    x

Tactic:
ext