awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Multicoequalizer (goal)
Goal:
X : Type u
S₁ S₂ S₃ S₄ : Set X
h : Lattice.BicartSq S₁ S₂ S₃ S₄
h₁ :
  Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R)
      {(MultispanShape.ofLinearOrder Bool).fst default, (MultispanShape.ofLinearOrder Bool).snd default} true →
    Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R) Set.univ true
h₂ :
  Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R) Set.univ true →
    Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R)
      {(MultispanShape.ofLinearOrder Bool).fst default, (MultispanShape.ofLinearOrder Bool).snd default} true
h₃ h₄ h₅ :
  Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R)
      {(MultispanShape.ofLinearOrder Bool).fst default, (MultispanShape.ofLinearOrder Bool).snd default} true →
    Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R) Set.univ true
h₆ h₇ :
  Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R) Set.univ true →
    Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R)
      {(MultispanShape.ofLinearOrder Bool).fst default, (MultispanShape.ofLinearOrder Bool).snd default} true
⊢ Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R)
      {(MultispanShape.ofLinearOrder Bool).fst default, (MultispanShape.ofLinearOrder Bool).snd default} true ↔
    Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R) Set.univ true
have h₈ : Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R) Set.univ true →
  ¬true = (MultispanShape.ofLinearOrder Bool).fst default →
    Membership.mem (γ := Set (MultispanShape.ofLinearOrder Bool).R) {(MultispanShape.ofLinearOrder Bool).snd default}
      true
Tactic: tauto
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Multiequalizer (goal)
Goal:
J : MulticospanShape
I : MulticospanIndex J (Type u)
s : (↑I.multicospan.sections : Type (max (max u u_2) u_1))
⊢ (fun (s : I.sections) ↦
        Subtype.mk (p := fun (x : (j : WalkingMulticospan J) → I.multicospan.obj j) ↦ x ∈ I.multicospan.sections)
          (fun (i : WalkingMulticospan J) ↦
            match (motive := (i : WalkingMulticospan J) → I.multicospan.obj i) i with
            | WalkingMulticospan.left i => s.val i
            | WalkingMulticospan.right j =>
              (ConcreteCategory.hom (FC := TypeCat.Fun) (I.fst j) :
                  (a : (fun (X : Type u) ↦ X) (I.left (J.fst j))) → (fun (X : Type u) ↦ X) (I.right j))
                (s.val (J.fst j)))
          ⋯)
      ((fun (s : (↑I.multicospan.sections : Type (max (max u u_2) u_1))) ↦
          ({ val := fun (i : J.L) ↦ (↑s : (j : WalkingMulticospan J) → I.multicospan.obj j) (WalkingMulticospan.left i),
              property := ⋯ } :
            I.sections))
        s) =
    s
have a : ∀ (a : J.L),
  (↑((fun (s : I.sections) ↦
              Subtype.mk (p := fun (x : (j : WalkingMulticospan J) → I.multicospan.obj j) ↦ x ∈ I.multicospan.sections)
                (fun (i : WalkingMulticospan J) ↦
                  match (motive := (i : WalkingMulticospan J) → I.multicospan.obj i) i with
                  | WalkingMulticospan.left i => s.val i
                  | WalkingMulticospan.right j =>
                    (ConcreteCategory.hom (FC := TypeCat.Fun) (I.fst j) :
                        (a : (fun (X : Type u) ↦ X) (I.left (J.fst j))) → (fun (X : Type u) ↦ X) (I.right j))
                      (s.val (J.fst j)))
                sorry)
            ((fun (s : (↑I.multicospan.sections : Type (max (max u u_2) u_1))) ↦
                ({
                    val := fun (i : J.L) ↦
                      (↑s : (j : WalkingMulticospan J) → I.multicospan.obj j) (WalkingMulticospan.left i),
                    property := sorry } :
                  I.sections))
              s)) :
        (j : WalkingMulticospan J) → I.multicospan.obj j)
      (WalkingMulticospan.left a) =
    (↑s : (j : WalkingMulticospan J) → I.multicospan.obj j) (WalkingMulticospan.left a)
Tactic: ext (_ | r)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Multiequalizer (goal)
Goal:
J : MulticospanShape
I : MulticospanIndex J (Type u)
c : Multifork I
⊢ Nonempty (IsLimit c) ↔ Function.Bijective c.toSections
have rw : Function.Bijective (Types.sectionOfCone c) ↔ Function.Bijective c.toSections
Tactic: rw [Types.isLimit_iff_bijective_sectionOfCone, ← toSections_fac, EquivLike.comp_bijective]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Multiequalizer (goal)
Goal:
J : MulticospanShape
I : MulticospanIndex J (Type u)
c : Multifork I
rw : Function.Bijective (Types.sectionOfCone c) ↔ Function.Bijective c.toSections
⊢ Nonempty (IsLimit c) ↔ Function.Bijective c.toSections
have rw₁ : Function.Bijective (Types.sectionOfCone c) ↔
  Function.Bijective (β := I.sections)
    ((⇑I.sectionsEquiv.symm : (a : (↑I.multicospan.sections : Type (max (max u u_1) u_2))) → I.sections) ∘
      Types.sectionOfCone c)
Tactic: rw [Types.isLimit_iff_bijective_sectionOfCone, ← toSections_fac, EquivLike.comp_bijective]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Multiequalizer (goal)
Goal:
J : MulticospanShape
I : MulticospanIndex J (Type u)
c : Multifork I
rw : Function.Bijective (Types.sectionOfCone c) ↔ Function.Bijective c.toSections
rw₁ :
  Function.Bijective (Types.sectionOfCone c) ↔
    Function.Bijective (β := I.sections)
      ((⇑I.sectionsEquiv.symm : (a : (↑I.multicospan.sections : Type (max (max u u_1) u_2))) → I.sections) ∘
        Types.sectionOfCone c)
⊢ Nonempty (IsLimit c) ↔ Function.Bijective c.toSections
have rw₂ : Function.Bijective (Types.sectionOfCone c) ↔ Function.Bijective (Types.sectionOfCone c)
Tactic: rw [Types.isLimit_iff_bijective_sectionOfCone, ← toSections_fac, EquivLike.comp_bijective]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Multiequalizer (goal)
Goal:
J : MulticospanShape
I : MulticospanIndex J (Type u)
c : Multifork I
hc : IsLimit c
s : I.sections
i : J.L
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι i) :
        (a : (fun (X : Type u) ↦ X) c.pt) → (fun (X : Type u) ↦ X) (I.left i))
      ((sectionsEquiv hc : (a : I.sections) → c.pt) s : c.pt) =
    s.val i
have obtain : ∀ (x : c.pt),
  Eq (α := (fun (X : Type u) ↦ X) (I.left i))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι i) :
          (a : (fun (X : Type u) ↦ X) c.pt) → (fun (X : Type u) ↦ X) (I.left i))
        ((sectionsEquiv hc : (a : I.sections) → c.pt) (((sectionsEquiv hc).symm : (a : c.pt) → I.sections) x) : c.pt) :
      (fun (X : Type u) ↦ X) (I.left i))
    (MulticospanIndex.sections.val (((sectionsEquiv hc).symm : (a : c.pt) → I.sections) x) i : I.left i)
Tactic: obtain ⟨x, rfl⟩ := (sectionsEquiv hc).symm.surjective s
Cleared hypotheses: s
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X Y : Type u
x : Cone (pair X Y)
x_1 : x.pt ⟶ (binaryProductCone X Y).pt
w : ∀ (j : Discrete WalkingPair), x_1 ≫ (binaryProductCone X Y).π.app j = x.π.app j
⊢ x_1 =
    ↾fun (x_2 : x.pt) ↦
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.fst x) :
            (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x.pt).obj { as := left })) →
              (fun (X : Type u) ↦ X) ((pair X Y).obj { as := left }))
          x_2,
        (ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.snd x) :
            (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x.pt).obj { as := right })) →
              (fun (X : Type u) ↦ X) ((pair X Y).obj { as := right }))
          x_2)
have h : ∀ (x_2 : x.pt),
  (ConcreteCategory.hom (FC := TypeCat.Fun) x_1).toFun x_2 =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (↾fun (x_3 : x.pt) ↦
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.fst x) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x.pt).obj { as := left })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := left }))
              x_3,
            (ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.snd x) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x.pt).obj { as := right })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := right }))
              x_3)))
      x_2
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X Y : Type u
x_1 : Cone (pair X Y)
x_2 : x_1.pt ⟶ (binaryProductCone X Y).pt
w : ∀ (j : Discrete WalkingPair), x_2 ≫ (binaryProductCone X Y).π.app j = x_1.π.app j
x : x_1.pt
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) x_2).toFun x =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (↾fun (x : x_1.pt) ↦
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.fst x_1) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := left })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := left }))
              x,
            (ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.snd x_1) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := right })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := right }))
              x)))
      x
have h : ((ConcreteCategory.hom (FC := TypeCat.Fun) x_2).toFun x).1 =
  (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (↾fun (x : x_1.pt) ↦
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.fst x_1) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := left })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := left }))
              x,
            (ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.snd x_1) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := right })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := right }))
              x)))
      x).1
Tactic: apply Prod.ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X Y : Type u
x_1 : Cone (pair X Y)
x_2 : x_1.pt ⟶ (binaryProductCone X Y).pt
w : ∀ (j : Discrete WalkingPair), x_2 ≫ (binaryProductCone X Y).π.app j = x_1.π.app j
x : x_1.pt
h :
  ((ConcreteCategory.hom (FC := TypeCat.Fun) x_2).toFun x).1 =
    (TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (↾fun (x : x_1.pt) ↦
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.fst x_1) :
                  (a :
                      (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := left })) →
                    (fun (X : Type u) ↦ X) ((pair X Y).obj { as := left }))
                x,
              (ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.snd x_1) :
                  (a :
                      (fun (X : Type u) ↦ X)
                        (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := right })) →
                    (fun (X : Type u) ↦ X) ((pair X Y).obj { as := right }))
                x)))
        x).1
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) x_2).toFun x =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (↾fun (x : x_1.pt) ↦
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.fst x_1) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := left })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := left }))
              x,
            (ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.snd x_1) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := right })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := right }))
              x)))
      x
have h₁ : ((ConcreteCategory.hom (FC := TypeCat.Fun) x_2).toFun x).2 =
  (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (↾fun (x : x_1.pt) ↦
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.fst x_1) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := left })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := left }))
              x,
            (ConcreteCategory.hom (FC := TypeCat.Fun) (BinaryFan.snd x_1) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const (Discrete WalkingPair)).obj x_1.pt).obj { as := right })) →
                  (fun (X : Type u) ↦ X) ((pair X Y).obj { as := right }))
              x)))
      x).2
Tactic: apply Prod.ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
⊢ binaryProductFunctor ≅ prod.functor
have refine_1 : (X : Type u) → binaryProductFunctor.obj X ≅ Functor.obj prod.functor X
Tactic: refine NatIso.ofComponents (fun X => ?_) (fun _ => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
refine_1 : (X : Type u) → binaryProductFunctor.obj X ≅ Functor.obj prod.functor X
⊢ binaryProductFunctor ≅ prod.functor
have refine_2 : ∀ {X Y : Type u} (x : X ⟶ Y),
  Eq (α := binaryProductFunctor.obj X ⟶ Functor.obj prod.functor Y)
    (binaryProductFunctor.map x ≫
      Iso.hom
        ((fun (X : Type u) ↦
            (NatIso.ofComponents
                (fun (Y : Type u) ↦
                  Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)))
                sorry :
              binaryProductFunctor.obj X ≅ Functor.obj prod.functor X))
          Y))
    (Iso.hom
        ((fun (X : Type u) ↦
            (NatIso.ofComponents
                (fun (Y : Type u) ↦
                  Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)))
                sorry :
              binaryProductFunctor.obj X ≅ Functor.obj prod.functor X))
          X) ≫
      Functor.map prod.functor x)
Tactic: refine NatIso.ofComponents (fun X => ?_) (fun _ => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X : Type u
⊢ binaryProductFunctor.obj X ≅ Functor.obj prod.functor X
have refine_1 : (Y : Type u) → (binaryProductFunctor.obj X).obj Y ≅ Functor.obj (Functor.obj prod.functor X) Y
Tactic: refine NatIso.ofComponents (fun Y => ?_) (fun _ => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X : Type u
refine_1 : (Y : Type u) → (binaryProductFunctor.obj X).obj Y ≅ Functor.obj (Functor.obj prod.functor X) Y
⊢ binaryProductFunctor.obj X ≅ Functor.obj prod.functor X
have refine_1₁ : ∀ {X_1 Y : Type u} (x : X_1 ⟶ Y),
  Eq (α := (binaryProductFunctor.obj X).obj X_1 ⟶ Functor.obj (Functor.obj prod.functor X) Y)
    ((binaryProductFunctor.obj X).map x ≫
      Iso.hom
        ((fun (Y : Type u) ↦
            (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
              (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
          Y))
    (Iso.hom
        ((fun (Y : Type u) ↦
            (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
              (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
          X_1) ≫
      Functor.map (Functor.obj prod.functor X) x)
Tactic: refine NatIso.ofComponents (fun Y => ?_) (fun _ => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X X_1 Y : Type u
x : X_1 ⟶ Y
⊢ Eq (α := (binaryProductFunctor.obj X).obj X_1 ⟶ Functor.obj (Functor.obj prod.functor X) Y)
    ((binaryProductFunctor.obj X).map x ≫
      Iso.hom
        ((fun (Y : Type u) ↦
            (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
              (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
          Y))
    (Iso.hom
        ((fun (Y : Type u) ↦
            (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
              (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
          X_1) ≫
      Functor.map (Functor.obj prod.functor X) x)
have refine_1 : Eq (α := (binaryProductFunctor.obj X).obj X_1 ⟶ X)
  (CategoryStruct.comp (Y := X ⨯ Y)
    ((binaryProductFunctor.obj X).map x ≫
      Iso.hom
        ((fun (Y : Type u) ↦
            (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
              (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
          Y))
    prod.fst)
  (CategoryStruct.comp (Y := X ⨯ Y)
    (Iso.hom
          ((fun (Y : Type u) ↦
              (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
                (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
            X_1) ≫
        Functor.map (Functor.obj prod.functor X) x :
      (binaryProductFunctor.obj X).obj X_1 ⟶ Functor.obj (Functor.obj prod.functor X) Y)
    prod.fst)
Tactic: apply Limits.prod.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X X_1 Y : Type u
x : X_1 ⟶ Y
refine_1 :
  Eq (α := (binaryProductFunctor.obj X).obj X_1 ⟶ X)
    (CategoryStruct.comp (Y := X ⨯ Y)
      ((binaryProductFunctor.obj X).map x ≫
        Iso.hom
          ((fun (Y : Type u) ↦
              (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
                (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
            Y))
      prod.fst)
    (CategoryStruct.comp (Y := X ⨯ Y)
      (Iso.hom
            ((fun (Y : Type u) ↦
                (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
                  (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
              X_1) ≫
          Functor.map (Functor.obj prod.functor X) x :
        (binaryProductFunctor.obj X).obj X_1 ⟶ Functor.obj (Functor.obj prod.functor X) Y)
      prod.fst)
⊢ Eq (α := (binaryProductFunctor.obj X).obj X_1 ⟶ Functor.obj (Functor.obj prod.functor X) Y)
    ((binaryProductFunctor.obj X).map x ≫
      Iso.hom
        ((fun (Y : Type u) ↦
            (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
              (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
          Y))
    (Iso.hom
        ((fun (Y : Type u) ↦
            (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
              (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
          X_1) ≫
      Functor.map (Functor.obj prod.functor X) x)
have refine_1₁ : Eq (α := (binaryProductFunctor.obj X).obj X_1 ⟶ Y)
  (CategoryStruct.comp (Y := X ⨯ Y)
    ((binaryProductFunctor.obj X).map x ≫
      Iso.hom
        ((fun (Y : Type u) ↦
            (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
              (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
          Y))
    prod.snd)
  (CategoryStruct.comp (Y := X ⨯ Y)
    (Iso.hom
          ((fun (Y : Type u) ↦
              (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
                (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
            X_1) ≫
        Functor.map (Functor.obj prod.functor X) x :
      (binaryProductFunctor.obj X).obj X_1 ⟶ Functor.obj (Functor.obj prod.functor X) Y)
    prod.snd)
Tactic: apply Limits.prod.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X X_1 Y : Type u
x : X_1 ⟶ Y
⊢ Eq (α := (binaryProductFunctor.obj X).obj X_1 ⟶ X)
    (CategoryStruct.comp (Y := X ⨯ Y)
      ((binaryProductFunctor.obj X).map x ≫
        Iso.hom
          ((fun (Y : Type u) ↦
              (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
                (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
            Y))
      prod.fst)
    (CategoryStruct.comp (Y := X ⨯ Y)
      (Iso.hom
            ((fun (Y : Type u) ↦
                (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
                  (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
              X_1) ≫
          Functor.map (Functor.obj prod.functor X) x :
        (binaryProductFunctor.obj X).obj X_1 ⟶ Functor.obj (Functor.obj prod.functor X) Y)
      prod.fst)
have refine_1 : Eq.{u + 1} (α := X × X_1 ⟶ X)
  (CategoryStruct.comp (Y := X × Y)
    (↾fun (x_1 : X × X_1) ↦ (x_1.1, (ConcreteCategory.hom (FC := TypeCat.Fun) x : (a : X_1) → Y) x_1.2))
    (↾_root_.Prod.fst))
  (↾_root_.Prod.fst)
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X X_1 Y : Type u
x : X_1 ⟶ Y
⊢ Eq (α := (binaryProductFunctor.obj X).obj X_1 ⟶ Y)
    (CategoryStruct.comp (Y := X ⨯ Y)
      ((binaryProductFunctor.obj X).map x ≫
        Iso.hom
          ((fun (Y : Type u) ↦
              (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
                (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
            Y))
      prod.snd)
    (CategoryStruct.comp (Y := X ⨯ Y)
      (Iso.hom
            ((fun (Y : Type u) ↦
                (Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)) :
                  (binaryProductCone X Y).pt ≅ Cone.pt (F := pair X Y) (limit.cone (pair X Y))))
              X_1) ≫
          Functor.map (Functor.obj prod.functor X) x :
        (binaryProductFunctor.obj X).obj X_1 ⟶ Functor.obj (Functor.obj prod.functor X) Y)
      prod.snd)
have refine_1 : Eq (α := X × X_1 ⟶ Y)
  (CategoryStruct.comp (Y := X × Y)
    (↾fun (x_1 : X × X_1) ↦ (x_1.1, (ConcreteCategory.hom (FC := TypeCat.Fun) x : (a : X_1) → Y) x_1.2))
    (↾_root_.Prod.snd))
  (↾_root_.Prod.snd ≫ x)
Tactic: simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Products (goal)
Goal:
X Y : Type u
x : X ⟶ Y
⊢ Eq (α := binaryProductFunctor.obj X ⟶ Functor.obj prod.functor Y)
    (binaryProductFunctor.map x ≫
      Iso.hom
        ((fun (X : Type u) ↦
            (NatIso.ofComponents
                (fun (Y : Type u) ↦
                  Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)))
                ⋯ :
              binaryProductFunctor.obj X ≅ Functor.obj prod.functor X))
          Y))
    (Iso.hom
        ((fun (X : Type u) ↦
            (NatIso.ofComponents
                (fun (Y : Type u) ↦
                  Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)))
                ⋯ :
              binaryProductFunctor.obj X ≅ Functor.obj prod.functor X))
          X) ≫
      Functor.map prod.functor x)
have refine_2 : ∀ (x_1 : Type u),
  Eq (α := (binaryProductFunctor.obj X).obj x_1 ⟶ Functor.obj (Functor.obj prod.functor Y) x_1)
    (NatTrans.app
      (binaryProductFunctor.map x ≫
        Iso.hom
          ((fun (X : Type u) ↦
              (NatIso.ofComponents
                  (fun (Y : Type u) ↦
                    Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)))
                  sorry :
                binaryProductFunctor.obj X ≅ Functor.obj prod.functor X))
            Y))
      x_1)
    (NatTrans.app
      (Iso.hom
          ((fun (X : Type u) ↦
              (NatIso.ofComponents
                  (fun (Y : Type u) ↦
                    Iso.symm (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)))
                  sorry :
                binaryProductFunctor.obj X ≅ Functor.obj prod.functor X))
            X) ≫
        Functor.map prod.functor x)
      x_1)
Tactic: ext : 2