awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
ι : Type v
α : ι → TopCat
S : Cone (Discrete.functor α)
m : S.pt ⟶ (piFan.{v, u} α).pt
h : ∀ (j : Discrete ι), m ≫ (piFan.{v, u} α).π.app j = S.π.app j
x : (↑S.pt : Type (max u v))
i : ι
⊢ Eq (α := (↑(α i) : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
            m :
          (a : (↑S.pt : Type (max u v))) → (↑(piFan α).pt : Type (max u v)))
        x i :
      (↑(α i) : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
            (X := S.pt) (Y := (piFan.{v, u} α).pt)
            (ofHom
              {
                toFun := fun (s : (↑S.1 : Type (max u v))) (i : ι) ↦
                  (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v))))
                        (CC := carrier) (X :=
                        Functor.obj (Functor.obj (Functor.const (Discrete ι)) S.pt) { as := i : Discrete ι })
                        (S.π.app { as := i }) :
                      (a : (↑(((Functor.const (Discrete ι)).obj S.pt).obj { as := i }) : Type (max u v))) →
                        (↑((Discrete.functor α).obj { as := i }) : Type (max u v)))
                    s,
                continuous_toFun := ⋯ }) :
          (a : (↑S.pt : Type (max u v))) → (↑(piFan α).pt : Type (max u v)))
        x i :
      (↑(α i) : Type (max u v)))

Tactic:
simp [ContinuousMap.coe_mk, ← h ⟨i⟩]

Premises used: CategoryTheory.Discrete.mk, ContinuousMap.coe_mk
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
ι : Type v
α : ι → TopCat
i : ι
⊢ (piIsoPi.{v, u} α).inv ≫ Pi.π α i = piπ.{v, u} α i

Tactic:
simp [piIsoPi]

Premises used: TopCat.piIsoPi
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
ι : Type v
α : ι → TopCat
i : ι
x : (i : ι) → (↑(α i) : Type (max u v))
⊢ (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
          (X := ∏ᶜ α) (Pi.π α i) :
        (a : (↑(∏ᶜ α) : Type (max u v))) → (↑(α i) : Type (max u v)))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC :=
            carrier) (piIsoPi.{v, u} α).inv :
          (a : (↑(of ((i : ι) → (↑(α i) : Type (max u v)))) : Type (max v u))) → (↑(∏ᶜ α) : Type (max v u)))
        x) =
    x i

Tactic:
ConcreteCategory.congr_hom (piIsoPi_inv_π α i) x

Premises used: TopCat.piIsoPi_inv_π, CategoryTheory.ConcreteCategory.congr_hom
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
ι : Type v
α : ι → TopCat
i : ι
x : (↑(∏ᶜ α) : Type (max u v))
⊢ Eq (α := (↑(α i) : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
            (piIsoPi.{v, u} α).hom :
          (a : (↑(∏ᶜ α) : Type (max v u))) → (↑(of ((i : ι) → (↑(α i) : Type (max u v)))) : Type (max v u)))
        x i :
      (↑(α i) : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
            (X := ∏ᶜ α) (Pi.π α i) :
          (a : (↑(∏ᶜ α) : Type (max u v))) → (↑(α i) : Type (max u v)))
        x :
      (↑(α i) : Type (max u v)))

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
ι : Type v
α : ι → TopCat
i : ι
refine_1 : (↑(α i) : Type (max u v)) → (i : ι) × (↑(α i) : Type (max u v))
refine_2 :
  Continuous (X := (↑(α i) : Type (max u v))) (Y := (i : ι) × (↑(α i) : Type (max u v)))
    (id (Sigma.mk (β := fun (i : ι) ↦ (↑(α i) : Type (max u v))) i))
⊢ α i ⟶ of ((i : ι) × (↑(α i) : Type (max u v)))

Tactic:
refine ofHom (ContinuousMap.mk ?_ ?_)

Premises used: ContinuousMap.mk, TopCat.ofHom
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
ι : Type v
α : ι → TopCat
i : ι
⊢ (↑(α i) : Type (max u v)) → (i : ι) × (↑(α i) : Type (max u v))

Tactic:
apply Sigma.mk i

Premises used: Sigma.mk
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
ι : Type v
α : ι → TopCat
i : ι
⊢ Continuous (X := (↑(α i) : Type (max u v))) (Y := (i : ι) × (↑(α i) : Type (max u v)))
    (Sigma.mk (β := fun (i : ι) ↦ (↑(α i) : Type (max u v))) i)

Tactic:
continuity
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
ι : Type v
β : ι → TopCat
S : Cocone (Discrete.functor β)
⊢ Continuous (Y := (↑S.1 : Type (max u v))) fun (s : (↑(of ((i : ι) × (↑(β i) : Type (max u v)))) : Type (max u v))) ↦
    (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
          (X :=
          Functor.obj (Discrete.functor β) { as := s.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι })
          (S.ι.app { as := s.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) }) :
        (a : (↑((Discrete.functor β).obj { as := s.fst }) : Type (max u v))) →
          (↑(((Functor.const (Discrete ι)).obj S.pt).obj { as := s.fst }) : Type (max u v)))
      (s.snd (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))))

Tactic:
continuity
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
ι : Type v
β : ι → TopCat
s : Cocone (Discrete.functor β)
j : Discrete ι
mk :
  ∀ (as : ι),
    Eq (α := Functor.obj (Discrete.functor β) { as := as : Discrete ι } ⟶ s.pt)
      ((sigmaCofan.{v, u} β).ι.app { as := as } ≫
        ofHom
          {
            toFun := fun (s_1 : (↑(of ((i : ι) × (↑(β i) : Type (max u v)))) : Type (max u v))) ↦
              (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                    carrier) (X :=
                    Functor.obj (Discrete.functor β)
                      { as := s_1.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι })
                    (s.ι.app { as := s_1.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι }) :
                  (a : (↑((Discrete.functor β).obj { as := s_1.fst }) : Type (max u v))) →
                    (↑(((Functor.const (Discrete ι)).obj s.pt).obj { as := s_1.fst }) : Type (max u v)))
                (s_1.snd (β := fun (i : ι) ↦ (↑(β i) : Type (max u v)))),
            continuous_toFun := ⋯ : C((i : ι) × (↑(β i) : Type (max u v)), (↑s.1 : Type (max u v))) })
      (s.ι.app { as := as })
⊢ (sigmaCofan.{v, u} β).ι.app j ≫
      ofHom
        {
          toFun := fun (s_1 : (↑(of ((i : ι) × (↑(β i) : Type (max u v)))) : Type (max u v))) ↦
            (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                  carrier) (X :=
                  Functor.obj (Discrete.functor β)
                    { as := s_1.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι })
                  (s.ι.app { as := s_1.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι }) :
                (a : (↑((Discrete.functor β).obj { as := s_1.fst }) : Type (max u v))) →
                  (↑(((Functor.const (Discrete ι)).obj s.pt).obj { as := s_1.fst }) : Type (max u v)))
              (s_1.snd (β := fun (i : ι) ↦ (↑(β i) : Type (max u v)))),
          continuous_toFun := ⋯ : C((i : ι) × (↑(β i) : Type (max u v)), (↑s.1 : Type (max u v))) } =
    s.ι.app j

Tactic:
cases j
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
ι : Type v
β : ι → TopCat
s : Cocone (Discrete.functor β)
as : ι
mk mk₁ mk₂ mk₃ :
  Eq (α := Functor.obj (Discrete.functor β) { as := as : Discrete ι } ⟶ s.pt)
    ((sigmaCofan.{v, u} β).ι.app { as := as } ≫
      ofHom
        {
          toFun := fun (s_1 : (↑(of ((i : ι) × (↑(β i) : Type (max u v)))) : Type (max u v))) ↦
            (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                  carrier) (X :=
                  Functor.obj (Discrete.functor β)
                    { as := s_1.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι })
                  (s.ι.app { as := s_1.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι }) :
                (a : (↑((Discrete.functor β).obj { as := s_1.fst }) : Type (max u v))) →
                  (↑(((Functor.const (Discrete ι)).obj s.pt).obj { as := s_1.fst }) : Type (max u v)))
              (s_1.snd (β := fun (i : ι) ↦ (↑(β i) : Type (max u v)))),
          continuous_toFun := ⋯ : C((i : ι) × (↑(β i) : Type (max u v)), (↑s.1 : Type (max u v))) })
    (s.ι.app { as := as })
⊢ Eq (α := Functor.obj (Discrete.functor β) { as := as : Discrete ι } ⟶ s.pt)
    ((sigmaCofan.{v, u} β).ι.app { as := as } ≫
      ofHom
        {
          toFun := fun (s_1 : (↑(of ((i : ι) × (↑(β i) : Type (max u v)))) : Type (max u v))) ↦
            (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                  carrier) (X :=
                  Functor.obj (Discrete.functor β)
                    { as := s_1.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι })
                  (s.ι.app { as := s_1.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι }) :
                (a : (↑((Discrete.functor β).obj { as := s_1.fst }) : Type (max u v))) →
                  (↑(((Functor.const (Discrete ι)).obj s.pt).obj { as := s_1.fst }) : Type (max u v)))
              (s_1.snd (β := fun (i : ι) ↦ (↑(β i) : Type (max u v)))),
          continuous_toFun := ⋯ : C((i : ι) × (↑(β i) : Type (max u v)), (↑s.1 : Type (max u v))) })
    (s.ι.app { as := as })

Tactic:
cat_disch
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
ι : Type v
β : ι → TopCat
S : Cocone (Discrete.functor β)
m : (sigmaCofan.{v, u} β).pt ⟶ S.pt
h : ∀ (j : Discrete ι), (sigmaCofan.{v, u} β).ι.app j ≫ m = S.ι.app j
w :
  ∀ (i : ι) (x : (↑(β i) : Type (max u v))),
    Eq (α := (↑S.pt : Type (max u v)))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
              carrier) m :
            (a : (↑(sigmaCofan β).pt : Type (max u v))) → (↑S.pt : Type (max u v)))
          (Sigma.mk (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) i x) :
        (↑S.pt : Type (max u v)))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
              carrier)
              (ofHom
                {
                  toFun := fun (s : (↑(of ((i : ι) × (↑(β i) : Type (max u v)))) : Type (max u v))) ↦
                    (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v))))
                          (CC := carrier) (X :=
                          Functor.obj (Discrete.functor β)
                            { as := s.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι })
                          (S.ι.app { as := s.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι }) :
                        (a : (↑((Discrete.functor β).obj { as := s.fst }) : Type (max u v))) →
                          (↑(((Functor.const (Discrete ι)).obj S.pt).obj { as := s.fst }) : Type (max u v)))
                      (s.snd (β := fun (i : ι) ↦ (↑(β i) : Type (max u v)))),
                  continuous_toFun := ⋯ : C((i : ι) × (↑(β i) : Type (max u v)), (↑S.1 : Type (max u v))) }) :
            (a : (↑(sigmaCofan β).pt : Type (max u v))) → (↑S.pt : Type (max u v)))
          (Sigma.mk (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) i x) :
        (↑S.pt : Type (max u v)))
⊢ m =
    ofHom
      {
        toFun := fun (s : (↑(of ((i : ι) × (↑(β i) : Type (max u v)))) : Type (max u v))) ↦
          (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                carrier) (X :=
                Functor.obj (Discrete.functor β)
                  { as := s.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι })
                (S.ι.app { as := s.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) }) :
              (a : (↑((Discrete.functor β).obj { as := s.fst }) : Type (max u v))) →
                (↑(((Functor.const (Discrete ι)).obj S.pt).obj { as := s.fst }) : Type (max u v)))
            (s.snd (β := fun (i : ι) ↦ (↑(β i) : Type (max u v)))),
        continuous_toFun := ⋯ }

Tactic:
ext ⟨i, x⟩
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
ι : Type v
β : ι → TopCat
S : Cocone (Discrete.functor β)
m : (sigmaCofan.{v, u} β).pt ⟶ S.pt
h : ∀ (j : Discrete ι), (sigmaCofan.{v, u} β).ι.app j ≫ m = S.ι.app j
i : ι
x : (↑(β i) : Type (max u v))
⊢ Eq (α := (↑S.pt : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
            m :
          (a : (↑(sigmaCofan β).pt : Type (max u v))) → (↑S.pt : Type (max u v)))
        (Sigma.mk (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) i x) :
      (↑S.pt : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
            (ofHom
              {
                toFun := fun (s : (i : ι) × (↑(β i) : Type (max u v))) ↦
                  (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v))))
                        (CC := carrier) (X :=
                        Functor.obj (Discrete.functor β)
                          { as := s.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι })
                        ((sigmaCofan.{v, u} β).ι.app
                            { as := s.fst (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) : Discrete ι } ≫
                          m) :
                      (a : (↑((Discrete.functor β).obj { as := s.fst }) : Type (max u v))) →
                        (↑(((Functor.const (Discrete ι)).obj S.pt).obj { as := s.fst }) : Type (max u v)))
                    (s.snd (β := fun (i : ι) ↦ (↑(β i) : Type (max u v)))),
                continuous_toFun := ⋯ : C((i : ι) × (↑(β i) : Type (max u v)), (↑S.1 : Type (max u v))) }) :
          (a : (↑(sigmaCofan β).pt : Type (max u v))) → (↑S.pt : Type (max u v)))
        (Sigma.mk (β := fun (i : ι) ↦ (↑(β i) : Type (max u v))) i x) :
      (↑S.pt : Type (max u v)))

Tactic:
congr
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
ι : Type v
α : ι → TopCat
i : ι
⊢ Sigma.ι α i ≫ (sigmaIsoSigma.{v, u} α).hom = sigmaι.{v, u} α i

Tactic:
simp [sigmaIsoSigma]

Premises used: TopCat.sigmaIsoSigma
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
ι : Type v
α : ι → TopCat
i : ι
x : (↑(α i) : Type (max u v))
⊢ Eq (α := (↑(of ((i : ι) × (↑(α i) : Type (max u v)))) : Type (max v u)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
            (sigmaIsoSigma.{v, u} α).hom :
          (a : (↑(∐ α) : Type (max v u))) → (↑(of ((i : ι) × (↑(α i) : Type (max u v)))) : Type (max v u)))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
              carrier) (Y := ∐ α) (Sigma.ι α i) :
            (a : (↑(α i) : Type (max u v))) → (↑(∐ α) : Type (max u v)))
          x) :
      (↑(of ((i : ι) × (↑(α i) : Type (max u v)))) : Type (max v u)))
    (Sigma.mk (β := fun (i : ι) ↦ (↑(α i) : Type (max u v))) i x : (i : ι) × (↑(α i) : Type (max u v)))

Tactic:
ConcreteCategory.congr_hom (sigmaIsoSigma_hom_ι α i) x

Premises used: TopCat.sigmaIsoSigma_hom_ι, CategoryTheory.ConcreteCategory.congr_hom
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
X Y : TopCat
S : Cone (pair X Y)
left :
  Eq (α := S.pt ⟶ X)
    (ofHom
          {
            toFun := fun (s : (↑S.1 : Type u)) ↦
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                      (BinaryFan.fst S) :
                    (a : (↑S.pt : Type u)) → (↑X : Type u))
                  s,
                (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                      (BinaryFan.snd S) :
                    (a : (↑S.pt : Type u)) → (↑Y : Type u))
                  s),
            continuous_toFun := ⋯ : C((↑S.1 : Type u), (↑X : Type u) × (↑Y : Type u)) } ≫
        (X.prodBinaryFan Y).π.app { as := WalkingPair.left } :
      S.pt ⟶ X)
    (S.π.app { as := WalkingPair.left } :
      Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) S.pt)
          { as := WalkingPair.left : Discrete WalkingPair } ⟶
        Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair })
⊢ Eq (α := S.pt ⟶ Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair })
    (ofHom
        {
          toFun := fun (s : (↑S.1 : Type u)) ↦
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                    (BinaryFan.fst S) :
                  (a : (↑(((Functor.const (Discrete WalkingPair)).obj S.pt).obj { as := WalkingPair.left }) : Type u)) →
                    (↑((pair X Y).obj { as := WalkingPair.left }) : Type u))
                s,
              (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                    (BinaryFan.snd S) :
                  (a :
                      (↑(((Functor.const (Discrete WalkingPair)).obj S.pt).obj { as := WalkingPair.right }) : Type u)) →
                    (↑((pair X Y).obj { as := WalkingPair.right }) : Type u))
                s),
          continuous_toFun := ⋯ : C((↑S.1 : Type u), (↑X : Type u) × (↑Y : Type u)) } ≫
      (X.prodBinaryFan Y).π.app { as := WalkingPair.left })
    (S.π.app { as := WalkingPair.left })

Tactic:
dsimp
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
X Y : TopCat
S : Cone (pair X Y)
left :
  ∀ (x : (↑S.pt : Type u)),
    Eq (α := (↑X : Type u))
      (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y := X)
                (ofHom
                      {
                        toFun := fun (s : (↑S.1 : Type u)) ↦
                          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                                  carrier) (BinaryFan.fst S) :
                                (a : (↑S.pt : Type u)) → (↑X : Type u))
                              s,
                            (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u)))
                                  (CC := carrier) (BinaryFan.snd S) :
                                (a : (↑S.pt : Type u)) → (↑Y : Type u))
                              s),
                        continuous_toFun := ⋯ : C((↑S.1 : Type u), (↑X : Type u) × (↑Y : Type u)) } ≫
                    (X.prodBinaryFan Y).π.app { as := WalkingPair.left } :
                  S.pt ⟶ X) :
              C((↑S.pt : Type u), (↑X : Type u))) :
            (a : (↑S.pt : Type u)) → (↑X : Type u))
          x :
        (↑X : Type u))
      (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
                S.pt) (Y := X)
                (S.π.app { as := WalkingPair.left } :
                  Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) S.pt)
                      { as := WalkingPair.left : Discrete WalkingPair } ⟶
                    Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) :
              C((↑S.pt : Type u), (↑X : Type u))) :
            (a : (↑S.pt : Type u)) → (↑X : Type u))
          x :
        (↑X : Type u))
⊢ Eq (α := S.pt ⟶ X)
    (ofHom
          {
            toFun := fun (s : (↑S.1 : Type u)) ↦
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                      (BinaryFan.fst S) :
                    (a : (↑S.pt : Type u)) → (↑X : Type u))
                  s,
                (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                      (BinaryFan.snd S) :
                    (a : (↑S.pt : Type u)) → (↑Y : Type u))
                  s),
            continuous_toFun := ⋯ : C((↑S.1 : Type u), (↑X : Type u) × (↑Y : Type u)) } ≫
        (X.prodBinaryFan Y).π.app { as := WalkingPair.left } :
      S.pt ⟶ X)
    (S.π.app { as := WalkingPair.left } :
      Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) S.pt)
          { as := WalkingPair.left : Discrete WalkingPair } ⟶
        Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair })

Tactic:
ext
-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
J : Type v
inst : Category.{w, v} J
X Y : TopCat
S : Cone (pair X Y)
x : (↑S.pt : Type u)
⊢ Eq (α := (↑X : Type u))
    (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y := X)
              (ofHom
                    {
                      toFun := fun (s : (↑S.1 : Type u)) ↦
                        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                                carrier) (BinaryFan.fst S) :
                              (a : (↑S.pt : Type u)) → (↑X : Type u))
                            s,
                          (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                                carrier) (BinaryFan.snd S) :
                              (a : (↑S.pt : Type u)) → (↑Y : Type u))
                            s),
                      continuous_toFun := ⋯ : C((↑S.1 : Type u), (↑X : Type u) × (↑Y : Type u)) } ≫
                  (X.prodBinaryFan Y).π.app { as := WalkingPair.left } :
                S.pt ⟶ X) :
            C((↑S.pt : Type u), (↑X : Type u))) :
          (a : (↑S.pt : Type u)) → (↑X : Type u))
        x :
      (↑X : Type u))
    (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X := S.pt)
              (Y := X)
              (S.π.app { as := WalkingPair.left } :
                Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) S.pt)
                    { as := WalkingPair.left : Discrete WalkingPair } ⟶
                  Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) :
            C((↑S.pt : Type u), (↑X : Type u))) :
          (a : (↑S.pt : Type u)) → (↑X : Type u))
        x :
      (↑X : Type u))

Tactic:
rfl