l3lab/ntp-mathlib huggingface dataset:
/-- The inclusion to the coproduct as a bundled continuous map. -/
abbrev sigmaι {ι : Type v} (α : ι → TopCatMax.{v,u}) (i : ι) : α i ⟶ TopCat.of (Σi, α i) := by
  refine ContinuousMap.mk ?_ ?_
  · dsimp
    apply Sigma.mk i
  · 

Proof state:
case refine_2
J : Type v
inst✝ : SmallCategory J
ι : Type v
α : ι → TopCatMax
i : ι
⊢ Continuous (id (Sigma.mk i))

Next tactic:
dsimp
/-- The inclusion to the coproduct as a bundled continuous map. -/
abbrev sigmaι {ι : Type v} (α : ι → TopCatMax.{v,u}) (i : ι) : α i ⟶ TopCat.of (Σi, α i) := by
  refine ContinuousMap.mk ?_ ?_
  · dsimp
    apply Sigma.mk i
  · dsimp; 

Proof state:
case refine_2
J : Type v
inst✝ : SmallCategory J
ι : Type v
α : ι → TopCatMax
i : ι
⊢ Continuous (Sigma.mk i)

Next tactic:
continuity
/-- The constructed cofan is indeed a colimit -/
def sigmaCofanIsColimit {ι : Type v} (β : ι → TopCatMax.{v, u}) : IsColimit (sigmaCofan β) where
  desc S :=
    { toFun := fun (s : of (Σ i, β i)) => S.ι.app ⟨s.1⟩ s.2
      continuous_toFun := continuous_sigma fun i => (S.ι.app ⟨i⟩).continuous_toFun }
  uniq := by
    intro S m h
    ext ⟨i, x⟩
    simp only [comp_app,hom_apply,← h ⟨i⟩]
    congr
  fac s j := by
    

Proof state:
J : Type v
inst✝ : SmallCategory J
ι : Type v
β : ι → TopCatMax
s : Cocone (Discrete.functor β)
j : Discrete ι
⊢ (sigmaCofan β).ι.app j ≫ (fun S => ContinuousMap.mk fun s => (S.ι.app { as := s.fst }) s.snd) s = s.ι.app j

Next tactic:
cases j
/-- The constructed cofan is indeed a colimit -/
def sigmaCofanIsColimit {ι : Type v} (β : ι → TopCatMax.{v, u}) : IsColimit (sigmaCofan β) where
  desc S :=
    { toFun := fun (s : of (Σ i, β i)) => S.ι.app ⟨s.1⟩ s.2
      continuous_toFun := continuous_sigma fun i => (S.ι.app ⟨i⟩).continuous_toFun }
  uniq := by
    intro S m h
    ext ⟨i, x⟩
    simp only [comp_app,hom_apply,← h ⟨i⟩]
    congr
  fac s j := by
    cases j
    

Proof state:
case mk
J : Type v
inst✝ : SmallCategory J
ι : Type v
β : ι → TopCatMax
s : Cocone (Discrete.functor β)
as✝ : ι
⊢ (sigmaCofan β).ι.app { as := as✝ } ≫ (fun S => ContinuousMap.mk fun s => (S.ι.app { as := s.fst }) s.snd) s =
    s.ι.app { as := as✝ }

Next tactic:
aesop_cat
/-- The constructed cofan is indeed a colimit -/
def sigmaCofanIsColimit {ι : Type v} (β : ι → TopCatMax.{v, u}) : IsColimit (sigmaCofan β) where
  desc S :=
    { toFun := fun (s : of (Σ i, β i)) => S.ι.app ⟨s.1⟩ s.2
      continuous_toFun := continuous_sigma fun i => (S.ι.app ⟨i⟩).continuous_toFun }
  uniq := by
    

Proof state:
J : Type v
inst✝ : SmallCategory J
ι : Type v
β : ι → TopCatMax
⊢ ∀ (s : Cocone (Discrete.functor β)) (m : (sigmaCofan β).pt ⟶ s.pt),
    (∀ (j : Discrete ι), (sigmaCofan β).ι.app j ≫ m = s.ι.app j) →
      m = (fun S => ContinuousMap.mk fun s => (S.ι.app { as := s.fst }) s.snd) s

Next tactic:
intro S m h
/-- The constructed cofan is indeed a colimit -/
def sigmaCofanIsColimit {ι : Type v} (β : ι → TopCatMax.{v, u}) : IsColimit (sigmaCofan β) where
  desc S :=
    { toFun := fun (s : of (Σ i, β i)) => S.ι.app ⟨s.1⟩ s.2
      continuous_toFun := continuous_sigma fun i => (S.ι.app ⟨i⟩).continuous_toFun }
  uniq := by
    intro S m h
    

Proof state:
J : Type v
inst✝ : SmallCategory J
ι : Type v
β : ι → TopCatMax
S : Cocone (Discrete.functor β)
m : (sigmaCofan β).pt ⟶ S.pt
h : ∀ (j : Discrete ι), (sigmaCofan β).ι.app j ≫ m = S.ι.app j
⊢ m = (fun S => ContinuousMap.mk fun s => (S.ι.app { as := s.fst }) s.snd) S

Next tactic:
ext ⟨i, x⟩
/-- The constructed cofan is indeed a colimit -/
def sigmaCofanIsColimit {ι : Type v} (β : ι → TopCatMax.{v, u}) : IsColimit (sigmaCofan β) where
  desc S :=
    { toFun := fun (s : of (Σ i, β i)) => S.ι.app ⟨s.1⟩ s.2
      continuous_toFun := continuous_sigma fun i => (S.ι.app ⟨i⟩).continuous_toFun }
  uniq := by
    intro S m h
    ext ⟨i, x⟩
    

Proof state:
case w.mk
J : Type v
inst✝ : SmallCategory J
ι : Type v
β : ι → TopCatMax
S : Cocone (Discrete.functor β)
m : (sigmaCofan β).pt ⟶ S.pt
h : ∀ (j : Discrete ι), (sigmaCofan β).ι.app j ≫ m = S.ι.app j
i : ι
x : ↑(β i)
⊢ m { fst := i, snd := x } =
    ((fun S => ContinuousMap.mk fun s => (S.ι.app { as := s.fst }) s.snd) S) { fst := i, snd := x }

Next tactic:
simp only [comp_app,hom_apply,← h ⟨i⟩]
/-- The constructed cofan is indeed a colimit -/
def sigmaCofanIsColimit {ι : Type v} (β : ι → TopCatMax.{v, u}) : IsColimit (sigmaCofan β) where
  desc S :=
    { toFun := fun (s : of (Σ i, β i)) => S.ι.app ⟨s.1⟩ s.2
      continuous_toFun := continuous_sigma fun i => (S.ι.app ⟨i⟩).continuous_toFun }
  uniq := by
    intro S m h
    ext ⟨i, x⟩
    simp only [comp_app,hom_apply,← h ⟨i⟩]
    

Proof state:
case w.mk
J : Type v
inst✝ : SmallCategory J
ι : Type v
β : ι → TopCatMax
S : Cocone (Discrete.functor β)
m : (sigmaCofan β).pt ⟶ S.pt
h : ∀ (j : Discrete ι), (sigmaCofan β).ι.app j ≫ m = S.ι.app j
i : ι
x : ↑(β i)
⊢ ContinuousMap.toFun m { fst := i, snd := x } = ContinuousMap.toFun ((sigmaCofan β).ι.app { as := i } ≫ m) x

Next tactic:
congr
@[reassoc (attr := simp)]
theorem sigmaIsoSigma_hom_ι {ι : Type v} (α : ι → TopCatMax.{v, u}) (i : ι) :
    Sigma.ι α i ≫ (sigmaIsoSigma α).hom = sigmaι α i := by 

Proof state:
J : Type v
inst✝ : SmallCategory J
ι : Type v
α : ι → TopCatMax
i : ι
⊢ Sigma.ι α i ≫ (sigmaIsoSigma α).hom = sigmaι α i

Next tactic:
simp [sigmaIsoSigma]
@[simp]
theorem sigmaIsoSigma_inv_apply {ι : Type v} (α : ι → TopCatMax.{v, u}) (i : ι) (x : α i) :
    (sigmaIsoSigma α).inv ⟨i, x⟩ = (Sigma.ι α i : _) x := by
  

Proof state:
J : Type v
inst✝ : SmallCategory J
ι : Type v
α : ι → TopCatMax
i : ι
x : ↑(α i)
⊢ (sigmaIsoSigma α).inv { fst := i, snd := x } = (Sigma.ι α i) x

Next tactic:
rw [← sigmaIsoSigma_hom_ι_apply, ← comp_app, ← comp_app, Iso.hom_inv_id,
    Category.comp_id]
theorem induced_of_isLimit {F : J ⥤ TopCatMax.{v, u}} (C : Cone F) (hC : IsLimit C) :
    C.pt.str = ⨅ j, (F.obj j).str.induced (C.π.app j) := by
  

Proof state:
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
C : Cone F
hC : IsLimit C
⊢ C.pt.str = ⨅ j, induced (⇑(C.π.app j)) (F.obj j).str

Next tactic:
let homeo := homeoOfIso (hC.conePointUniqueUpToIso (limitConeInfiIsLimit F))
theorem induced_of_isLimit {F : J ⥤ TopCatMax.{v, u}} (C : Cone F) (hC : IsLimit C) :
    C.pt.str = ⨅ j, (F.obj j).str.induced (C.π.app j) := by
  let homeo := homeoOfIso (hC.conePointUniqueUpToIso (limitConeInfiIsLimit F))
  

Proof state:
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
C : Cone F
hC : IsLimit C
homeo : ↑C.pt ≃ₜ ↑(limitConeInfi F).pt := homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F))
⊢ C.pt.str = ⨅ j, induced (⇑(C.π.app j)) (F.obj j).str

Next tactic:
refine' homeo.inducing.induced.trans _
theorem induced_of_isLimit {F : J ⥤ TopCatMax.{v, u}} (C : Cone F) (hC : IsLimit C) :
    C.pt.str = ⨅ j, (F.obj j).str.induced (C.π.app j) := by
  let homeo := homeoOfIso (hC.conePointUniqueUpToIso (limitConeInfiIsLimit F))
  refine' homeo.inducing.induced.trans _
  

Proof state:
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
C : Cone F
hC : IsLimit C
homeo : ↑C.pt ≃ₜ ↑(limitConeInfi F).pt := homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F))
⊢ induced (⇑homeo) (topologicalSpace_coe (limitConeInfi F).pt) = ⨅ j, induced (⇑(C.π.app j)) (F.obj j).str

Next tactic:
change induced homeo (⨅ j : J, _) = _
theorem induced_of_isLimit {F : J ⥤ TopCatMax.{v, u}} (C : Cone F) (hC : IsLimit C) :
    C.pt.str = ⨅ j, (F.obj j).str.induced (C.π.app j) := by
  let homeo := homeoOfIso (hC.conePointUniqueUpToIso (limitConeInfiIsLimit F))
  refine' homeo.inducing.induced.trans _
  change induced homeo (⨅ j : J, _) = _
  

Proof state:
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
C : Cone F
hC : IsLimit C
homeo : ↑C.pt ≃ₜ ↑(limitConeInfi F).pt := homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F))
⊢ induced (⇑homeo) (⨅ j, induced ((Types.limitCone (F ⋙ forget TopCat)).π.app j) (F.obj j).str) =
    ⨅ j, induced (⇑(C.π.app j)) (F.obj j).str

Next tactic:
simp [induced_iInf, induced_compose]
theorem induced_of_isLimit {F : J ⥤ TopCatMax.{v, u}} (C : Cone F) (hC : IsLimit C) :
    C.pt.str = ⨅ j, (F.obj j).str.induced (C.π.app j) := by
  let homeo := homeoOfIso (hC.conePointUniqueUpToIso (limitConeInfiIsLimit F))
  refine' homeo.inducing.induced.trans _
  change induced homeo (⨅ j : J, _) = _
  simp [induced_iInf, induced_compose]
  

Proof state:
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
C : Cone F
hC : IsLimit C
homeo : ↑C.pt ≃ₜ ↑(limitConeInfi F).pt := homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F))
⊢ ⨅ i,
      induced
        ((Types.limitCone (F ⋙ forget TopCat)).π.app i ∘
          ⇑(homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F))))
        (F.obj i).str =
    ⨅ j, induced (⇑(C.π.app j)) (F.obj j).str

Next tactic:
rfl
/-- The first projection from the product. -/
abbrev prodFst {X Y : TopCat.{u}} : TopCat.of (X × Y) ⟶ X :=
  ⟨Prod.fst, by 

Proof state:
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
⊢ Continuous Prod.fst

Next tactic:
continuity
/-- The second projection from the product. -/
abbrev prodSnd {X Y : TopCat.{u}} : TopCat.of (X × Y) ⟶ Y :=
  ⟨Prod.snd, by 

Proof state:
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
⊢ Continuous Prod.snd

Next tactic:
continuity
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    

Proof state:
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
⊢ ∀ (s : Cone (pair X Y)) (j : Discrete WalkingPair),
    (fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) s ≫ (prodBinaryFan X Y).π.app j =
      s.π.app j

Next tactic:
rintro S (_ | _)
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> 

Proof state:
case mk.left
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
⊢ (fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S ≫
      (prodBinaryFan X Y).π.app { as := WalkingPair.left } =
    S.π.app { as := WalkingPair.left }

Next tactic:
{dsimp; ext; rfl}
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {

Proof state:
case mk.left
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
⊢ (fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S ≫
      (prodBinaryFan X Y).π.app { as := WalkingPair.left } =
    S.π.app { as := WalkingPair.left }

Next tactic:
dsimp
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; 

Proof state:
case mk.left
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
⊢ (ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) ≫ BinaryFan.fst (prodBinaryFan X Y) =
    BinaryFan.fst S

Next tactic:
ext
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; ext; 

Proof state:
case mk.left.w
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
x✝ : (forget TopCat).obj S.pt
⊢ ((ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) ≫ BinaryFan.fst (prodBinaryFan X Y)) x✝ =
    (BinaryFan.fst S) x✝

Next tactic:
rfl
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> 

Proof state:
case mk.right
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
⊢ (fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S ≫
      (prodBinaryFan X Y).π.app { as := WalkingPair.right } =
    S.π.app { as := WalkingPair.right }

Next tactic:
{dsimp; ext; rfl}
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {

Proof state:
case mk.right
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
⊢ (fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S ≫
      (prodBinaryFan X Y).π.app { as := WalkingPair.right } =
    S.π.app { as := WalkingPair.right }

Next tactic:
dsimp
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; 

Proof state:
case mk.right
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
⊢ (ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) ≫ BinaryFan.snd (prodBinaryFan X Y) =
    BinaryFan.snd S

Next tactic:
ext
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; ext; 

Proof state:
case mk.right.w
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
x✝ : (forget TopCat).obj S.pt
⊢ ((ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) ≫ BinaryFan.snd (prodBinaryFan X Y)) x✝ =
    (BinaryFan.snd S) x✝

Next tactic:
rfl
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; ext; rfl}
  uniq := by
    

Proof state:
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
⊢ ∀ (s : Cone (pair X Y)) (m : s.pt ⟶ (prodBinaryFan X Y).pt),
    (∀ (j : Discrete WalkingPair), m ≫ (prodBinaryFan X Y).π.app j = s.π.app j) →
      m = (fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) s

Next tactic:
intro S m h
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; ext; rfl}
  uniq := by
    intro S m h
    -- porting note: used to be `ext x`
    

Proof state:
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
m : S.pt ⟶ (prodBinaryFan X Y).pt
h : ∀ (j : Discrete WalkingPair), m ≫ (prodBinaryFan X Y).π.app j = S.π.app j
⊢ m = (fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S

Next tactic:
refine' ContinuousMap.ext (fun (x : ↥(S.pt)) => Prod.ext _ _)
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; ext; rfl}
  uniq := by
    intro S m h
    -- porting note: used to be `ext x`
    refine' ContinuousMap.ext (fun (x : ↥(S.pt)) => Prod.ext _ _)
    · 

Proof state:
case refine'_1
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
m : S.pt ⟶ (prodBinaryFan X Y).pt
h : ∀ (j : Discrete WalkingPair), m ≫ (prodBinaryFan X Y).π.app j = S.π.app j
x : ↑S.pt
⊢ (m x).1 = (((fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S) x).1

Next tactic:
specialize h ⟨WalkingPair.left⟩
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; ext; rfl}
  uniq := by
    intro S m h
    -- porting note: used to be `ext x`
    refine' ContinuousMap.ext (fun (x : ↥(S.pt)) => Prod.ext _ _)
    · specialize h ⟨WalkingPair.left⟩
      

Proof state:
case refine'_1
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
m : S.pt ⟶ (prodBinaryFan X Y).pt
x : ↑S.pt
h : m ≫ (prodBinaryFan X Y).π.app { as := WalkingPair.left } = S.π.app { as := WalkingPair.left }
⊢ (m x).1 = (((fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S) x).1

Next tactic:
apply_fun fun e => e x at h
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; ext; rfl}
  uniq := by
    intro S m h
    -- porting note: used to be `ext x`
    refine' ContinuousMap.ext (fun (x : ↥(S.pt)) => Prod.ext _ _)
    · specialize h ⟨WalkingPair.left⟩
      apply_fun fun e => e x at h
      

Proof state:
case refine'_1
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
m : S.pt ⟶ (prodBinaryFan X Y).pt
x : ↑S.pt
h : (m ≫ (prodBinaryFan X Y).π.app { as := WalkingPair.left }) x = (S.π.app { as := WalkingPair.left }) x
⊢ (m x).1 = (((fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S) x).1

Next tactic:
exact h
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; ext; rfl}
  uniq := by
    intro S m h
    -- porting note: used to be `ext x`
    refine' ContinuousMap.ext (fun (x : ↥(S.pt)) => Prod.ext _ _)
    · specialize h ⟨WalkingPair.left⟩
      apply_fun fun e => e x at h
      exact h
    · 

Proof state:
case refine'_2
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
m : S.pt ⟶ (prodBinaryFan X Y).pt
h : ∀ (j : Discrete WalkingPair), m ≫ (prodBinaryFan X Y).π.app j = S.π.app j
x : ↑S.pt
⊢ (m x).2 = (((fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S) x).2

Next tactic:
specialize h ⟨WalkingPair.right⟩
/-- The constructed binary fan is indeed a limit -/
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
  lift := fun S : BinaryFan X Y => {
    toFun := fun s => (S.fst s, S.snd s)
    -- Porting note: continuity failed again here. Lean cannot infer
    -- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
    continuous_toFun := (Continuous.prod_mk)
      (BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
  fac := by
    rintro S (_ | _) <;> {dsimp; ext; rfl}
  uniq := by
    intro S m h
    -- porting note: used to be `ext x`
    refine' ContinuousMap.ext (fun (x : ↥(S.pt)) => Prod.ext _ _)
    · specialize h ⟨WalkingPair.left⟩
      apply_fun fun e => e x at h
      exact h
    · specialize h ⟨WalkingPair.right⟩
      

Proof state:
case refine'_2
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
S : Cone (pair X Y)
m : S.pt ⟶ (prodBinaryFan X Y).pt
x : ↑S.pt
h : m ≫ (prodBinaryFan X Y).π.app { as := WalkingPair.right } = S.π.app { as := WalkingPair.right }
⊢ (m x).2 = (((fun S => ContinuousMap.mk fun s => ((BinaryFan.fst S) s, (BinaryFan.snd S) s)) S) x).2

Next tactic:
apply_fun fun e => e x at h