awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
J : Type u_3
inst_2 : Category.{v_3, u_3} J
P : MorphismProperty.{v_1, u_1} T
inst_3 : P.RespectsIso
inst_4 : PreservesColimitsOfShape.{v_3, u_3, v_2, v_1, u_2, u_1} J L
inst_5 : HasColimitsOfShape.{v_3, u_3, v_2, u_2} J A
c : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [HasColimit D] → Cocone D
hc : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [inst_6 : HasColimit D] → IsColimit (c D)
H :
  ∀ (D : Functor.{v_3, v_1, u_3, u_1} J T) [inst_6 : HasColimit D] {X : T} (s : D ⟶ Functor.obj (Functor.const J) X),
    (∀ (j : J), P (s.app j)) →
      P (X := (c D).pt) (Y := { pt := X, ι := s : Cocone D }.pt) (IsColimit.desc (hc D) { pt := X, ι := s })
X : T
Y : CostructuredArrow L X
d : ObjectProperty.ColimitOfShape (MorphismProperty.costructuredArrowObj L P) J Y
hd : IsColimit
  ((proj L X ⋙ L).mapCocone d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).cocone) :=
  isColimitOfPreserves (proj L X ⋙ L) d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).isColimit
heq :
  Y.hom =
    hd.desc
      { pt := X,
        ι :=
          {
            app := fun (j : J) ↦
              (d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).diag.obj j).hom,
            naturality := ⋯ } }
⊢ MorphismProperty.costructuredArrowObj L P Y
have rw : P (X :=
  Cocone.pt (F := d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).diag ⋙ proj L X ⋙ L)
    (c (d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).diag ⋙ proj L X ⋙ L)))
  (Y := X)
  (IsColimit.desc (hc (d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).diag ⋙ proj L X ⋙ L))
      { pt := X,
        ι :=
          {
            app := fun (j : J) ↦
              (d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).diag.obj j).hom,
            naturality := sorry } } :
    Cocone.pt (F := d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).diag ⋙ proj L X ⋙ L)
        (c (d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).diag ⋙ proj L X ⋙ L)) ⟶
      Cocone.pt.{v_3, v_1, u_3, u_1} (J := J) (F :=
        d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).diag ⋙ proj L X ⋙ L)
        { pt := X,
          ι :=
            {
              app := fun (j : J) ↦
                (d.toColimitPresentation (P := MorphismProperty.costructuredArrowObj L P).diag.obj j).hom,
              naturality := sorry } })
Tactic: rw [P.costructuredArrowObj_iff, heq, ← hd.coconePointUniqueUpToIso_hom_desc (hc _),
      P.cancel_left_of_respectsIso]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (hypothesis)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
inst_2 : HasPullbacks.{v_2, u_2} A
inst_3 : HasPullbacks.{v_1, u_1} T
inst_4 : PreservesLimitsOfShape WalkingCospan L
X : T
inst_5 : P.IsStableUnderComposition
inst_6 : P.IsStableUnderBaseChange
inst_7 : P.HasOfPostcompProperty P
Y : CostructuredArrow L X
pres : LimitPresentation WalkingCospan Y
hpres : ∀ (j : WalkingCospan), MorphismProperty.costructuredArrowObj L P (pres.diag.obj j)
⊢ MorphismProperty.costructuredArrowObj L P Y
have h : IsPullback (L.map (pres.π.app WalkingCospan.left).left) (L.map (pres.π.app WalkingCospan.right).left)
  (L.map (pres.diag.map WalkingCospan.Hom.inl).left) (L.map (pres.diag.map WalkingCospan.Hom.inr).left)
Tactic: have h : IsPullback (L.map (pres.π.app .left).left) (L.map (pres.π.app .right).left)
        (L.map (pres.diag.map WalkingCospan.Hom.inl).left)
          (L.map (pres.diag.map WalkingCospan.Hom.inr).left) :=
      IsPullback.of_isLimit_cone <| isLimitOfPreserves
        (CategoryTheory.CostructuredArrow.toOver L X ⋙ CategoryTheory.Over.forget X) pres.isLimit
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
inst_2 : HasPullbacks.{v_2, u_2} A
inst_3 : HasPullbacks.{v_1, u_1} T
inst_4 : PreservesLimitsOfShape WalkingCospan L
X : T
inst_5 : P.IsStableUnderComposition
inst_6 : P.IsStableUnderBaseChange
inst_7 : P.HasOfPostcompProperty P
Y : CostructuredArrow L X
pres : LimitPresentation WalkingCospan Y
hpres : ∀ (j : WalkingCospan), MorphismProperty.costructuredArrowObj L P (pres.diag.obj j)
h :
  IsPullback (L.map (pres.π.app WalkingCospan.left).left) (L.map (pres.π.app WalkingCospan.right).left)
    (L.map (pres.diag.map WalkingCospan.Hom.inl).left) (L.map (pres.diag.map WalkingCospan.Hom.inr).left)
⊢ MorphismProperty.costructuredArrowObj L P Y
have rw : P Y.hom
Tactic: rw [MorphismProperty.costructuredArrowObj_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
inst_2 : HasPullbacks.{v_2, u_2} A
inst_3 : HasPullbacks.{v_1, u_1} T
inst_4 : PreservesLimitsOfShape WalkingCospan L
X : T
inst_5 : P.IsStableUnderComposition
inst_6 : P.IsStableUnderBaseChange
inst_7 : P.HasOfPostcompProperty P
Y : CostructuredArrow L X
pres : LimitPresentation WalkingCospan Y
hpres : ∀ (j : WalkingCospan), MorphismProperty.costructuredArrowObj L P (pres.diag.obj j)
h :
  IsPullback (L.map (pres.π.app WalkingCospan.left).left) (L.map (pres.π.app WalkingCospan.right).left)
    (L.map (pres.diag.map WalkingCospan.Hom.inl).left) (L.map (pres.diag.map WalkingCospan.Hom.inr).left)
⊢ P Y.hom
have rw : P (L.map (pres.π.app WalkingCospan.left).left ≫ (pres.diag.obj WalkingCospan.left).hom)
Tactic: rw [show Y.hom = L.map (pres.π.app .left).left ≫ (pres.diag.obj .left).hom by simp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
inst_2 : HasPullbacks.{v_2, u_2} A
inst_3 : HasPullbacks.{v_1, u_1} T
inst_4 : PreservesLimitsOfShape WalkingCospan L
X : T
inst_5 : P.IsStableUnderComposition
inst_6 : P.IsStableUnderBaseChange
inst_7 : P.HasOfPostcompProperty P
Y : CostructuredArrow L X
pres : LimitPresentation WalkingCospan Y
hpres : ∀ (j : WalkingCospan), MorphismProperty.costructuredArrowObj L P (pres.diag.obj j)
h :
  IsPullback (L.map (pres.π.app WalkingCospan.left).left) (L.map (pres.π.app WalkingCospan.right).left)
    (L.map (pres.diag.map WalkingCospan.Hom.inl).left) (L.map (pres.diag.map WalkingCospan.Hom.inr).left)
⊢ P (L.map (pres.π.app WalkingCospan.left).left ≫ (pres.diag.obj WalkingCospan.left).hom)
have apply : P (L.map (pres.diag.map WalkingCospan.Hom.inr).left)
Tactic: apply P.comp_mem _ _ (P.of_isPullback h.flip ?_) (hpres _)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
X : T
inst_2 : P.IsStableUnderComposition
inst_3 : P.IsStableUnderBaseChange
inst_4 : P.HasOfPostcompProperty P
inst_5 : HasPullbacks.{v_2, u_2} A
inst_6 : HasPullbacks.{v_1, u_1} T
inst_7 : PreservesLimitsOfShape WalkingCospan L
⊢ CreatesLimitsOfShape (C := P.CostructuredArrow ⊤ L X) WalkingCospan (CostructuredArrow.forget P ⊤ L X)
have inst : HasLimitsOfShape WalkingCospan (Comma L (Functor.fromPUnit X))
Tactic: apply +allowSynthFailures forgetCreatesLimitsOfShapeOfClosed
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst_1 : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
A : Type u_2
inst_2 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
X : T
inst_3 : P.IsStableUnderComposition
inst_4 : P.IsStableUnderBaseChange
inst_5 : P.HasOfPostcompProperty P
inst_6 : HasPullbacks.{v_2, u_2} A
inst_7 : HasPullbacks.{v_1, u_1} T
inst_8 : PreservesLimitsOfShape WalkingCospan L
inst : HasLimitsOfShape WalkingCospan (Comma L (Functor.fromPUnit X))
⊢ CreatesLimitsOfShape (C := P.CostructuredArrow ⊤ L X) WalkingCospan (CostructuredArrow.forget P ⊤ L X)
have inst₁ : (commaObj L (Functor.fromPUnit X) P).IsClosedUnderLimitsOfShape WalkingCospan
Tactic: apply +allowSynthFailures forgetCreatesLimitsOfShapeOfClosed
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
X : T
inst_2 : P.IsStableUnderComposition
inst_3 : P.IsStableUnderBaseChange
inst_4 : P.HasOfPostcompProperty P
inst_5 : HasPullbacks.{v_2, u_2} A
inst_6 : HasPullbacks.{v_1, u_1} T
inst_7 : PreservesLimitsOfShape WalkingCospan L
⊢ HasPullbacks (P.CostructuredArrow ⊤ L X)
have inst : HasLimitsOfShape WalkingCospan (Comma L (Functor.fromPUnit X))
Tactic: apply +allowSynthFailures hasLimitsOfShape_of_closedUnderLimitsOfShape
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst_1 : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
A : Type u_2
inst_2 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
X : T
inst_3 : P.IsStableUnderComposition
inst_4 : P.IsStableUnderBaseChange
inst_5 : P.HasOfPostcompProperty P
inst_6 : HasPullbacks.{v_2, u_2} A
inst_7 : HasPullbacks.{v_1, u_1} T
inst_8 : PreservesLimitsOfShape WalkingCospan L
inst : HasLimitsOfShape WalkingCospan (Comma L (Functor.fromPUnit X))
⊢ HasPullbacks (P.CostructuredArrow ⊤ L X)
have inst₁ : (commaObj L (Functor.fromPUnit X) P).IsClosedUnderLimitsOfShape WalkingCospan
Tactic: apply +allowSynthFailures hasLimitsOfShape_of_closedUnderLimitsOfShape
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
J : Type u_3
inst_2 : Category.{v_3, u_3} J
P : MorphismProperty.{v_1, u_1} T
inst_3 : P.RespectsIso
inst_4 : PreservesLimitsOfShape.{v_3, u_3, v_2, v_1, u_2, u_1} J L
inst_5 : HasLimitsOfShape.{v_3, u_3, v_2, u_2} J A
c : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [HasLimit D] → Cone D
hc : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [inst_6 : HasLimit D] → IsLimit (c D)
H :
  ∀ (D : Functor.{v_3, v_1, u_3, u_1} J T) [inst_6 : HasLimit D] {X : T} (s : Functor.obj (Functor.const J) X ⟶ D),
    (∀ (j : J), P (s.app j)) →
      P (X := { pt := X, π := s : Cone D }.pt) (Y := (c D).pt) (IsLimit.lift (hc D) { pt := X, π := s })
X : T
Y : StructuredArrow X L
d : ObjectProperty.LimitOfShape (MorphismProperty.structuredArrowObj L P) J Y
hd : IsLimit ((proj X L ⋙ L).mapCone d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).cone) :=
  isLimitOfPreserves (proj X L ⋙ L) d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).isLimit
⊢ Y.hom =
    hd.lift
      { pt := X,
        π :=
          { app := fun (j : J) ↦ (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag.obj j).hom,
            naturality := ⋯ } }
have refine : ∀ (j : J),
  Y.hom ≫ ((proj X L ⋙ L).mapCone d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).cone).π.app j =
    hd.lift
        { pt := X,
          π :=
            {
              app := fun (j : J) ↦
                (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag.obj j).hom,
              naturality := sorry } } ≫
      ((proj X L ⋙ L).mapCone d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).cone).π.app j
Tactic: refine hd.hom_ext fun j ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
J : Type u_3
inst_2 : Category.{v_3, u_3} J
P : MorphismProperty.{v_1, u_1} T
inst_3 : P.RespectsIso
inst_4 : PreservesLimitsOfShape.{v_3, u_3, v_2, v_1, u_2, u_1} J L
inst_5 : HasLimitsOfShape.{v_3, u_3, v_2, u_2} J A
c : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [HasLimit D] → Cone D
hc : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [inst_6 : HasLimit D] → IsLimit (c D)
H :
  ∀ (D : Functor.{v_3, v_1, u_3, u_1} J T) [inst_6 : HasLimit D] {X : T} (s : Functor.obj (Functor.const J) X ⟶ D),
    (∀ (j : J), P (s.app j)) →
      P (X := { pt := X, π := s : Cone D }.pt) (Y := (c D).pt) (IsLimit.lift (hc D) { pt := X, π := s })
X : T
Y : StructuredArrow X L
d : ObjectProperty.LimitOfShape (MorphismProperty.structuredArrowObj L P) J Y
hd : IsLimit ((proj X L ⋙ L).mapCone d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).cone) :=
  isLimitOfPreserves (proj X L ⋙ L) d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).isLimit
j : J
⊢ Y.hom ≫ ((proj X L ⋙ L).mapCone d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).cone).π.app j =
    hd.lift
        { pt := X,
          π :=
            {
              app := fun (j : J) ↦
                (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag.obj j).hom,
              naturality := ⋯ } } ≫
      ((proj X L ⋙ L).mapCone d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).cone).π.app j
have simp : Y.hom ≫ ((proj X L ⋙ L).mapCone d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).cone).π.app j =
  (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag.obj j).hom
Tactic: simp only [IsLimit.fac]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (hypothesis)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
J : Type u_3
inst_2 : Category.{v_3, u_3} J
P : MorphismProperty.{v_1, u_1} T
inst_3 : P.RespectsIso
inst_4 : PreservesLimitsOfShape.{v_3, u_3, v_2, v_1, u_2, u_1} J L
inst_5 : HasLimitsOfShape.{v_3, u_3, v_2, u_2} J A
c : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [HasLimit D] → Cone D
hc : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [inst_6 : HasLimit D] → IsLimit (c D)
H :
  ∀ (D : Functor.{v_3, v_1, u_3, u_1} J T) [inst_6 : HasLimit D] {X : T} (s : Functor.obj (Functor.const J) X ⟶ D),
    (∀ (j : J), P (s.app j)) →
      P (X := { pt := X, π := s : Cone D }.pt) (Y := (c D).pt) (IsLimit.lift (hc D) { pt := X, π := s })
X : T
Y : StructuredArrow X L
d : ObjectProperty.LimitOfShape (MorphismProperty.structuredArrowObj L P) J Y
hd : IsLimit ((proj X L ⋙ L).mapCone d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).cone) :=
  isLimitOfPreserves (proj X L ⋙ L) d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).isLimit
⊢ MorphismProperty.structuredArrowObj L P Y
have heq : Y.hom =
  hd.lift
    { pt := X,
      π :=
        { app := fun (j : J) ↦ (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag.obj j).hom,
          naturality := sorry } }
Tactic: have heq : Y.hom = hd.lift { pt := X, π := { app j := (d.diag.obj j).hom } } := by
      refine hd.hom_ext fun j ↦ ?_
      simp only [IsLimit.fac]
      simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
A : Type u_2
inst_1 : Category.{v_2, u_2} A
L : Functor.{v_2, v_1, u_2, u_1} A T
J : Type u_3
inst_2 : Category.{v_3, u_3} J
P : MorphismProperty.{v_1, u_1} T
inst_3 : P.RespectsIso
inst_4 : PreservesLimitsOfShape.{v_3, u_3, v_2, v_1, u_2, u_1} J L
inst_5 : HasLimitsOfShape.{v_3, u_3, v_2, u_2} J A
c : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [HasLimit D] → Cone D
hc : (D : Functor.{v_3, v_1, u_3, u_1} J T) → [inst_6 : HasLimit D] → IsLimit (c D)
H :
  ∀ (D : Functor.{v_3, v_1, u_3, u_1} J T) [inst_6 : HasLimit D] {X : T} (s : Functor.obj (Functor.const J) X ⟶ D),
    (∀ (j : J), P (s.app j)) →
      P (X := { pt := X, π := s : Cone D }.pt) (Y := (c D).pt) (IsLimit.lift (hc D) { pt := X, π := s })
X : T
Y : StructuredArrow X L
d : ObjectProperty.LimitOfShape (MorphismProperty.structuredArrowObj L P) J Y
hd : IsLimit ((proj X L ⋙ L).mapCone d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).cone) :=
  isLimitOfPreserves (proj X L ⋙ L) d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).isLimit
heq :
  Y.hom =
    hd.lift
      { pt := X,
        π :=
          { app := fun (j : J) ↦ (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag.obj j).hom,
            naturality := ⋯ } }
⊢ MorphismProperty.structuredArrowObj L P Y
have rw : P (X := X) (Y :=
  Cone.pt (F := d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag ⋙ proj X L ⋙ L)
    (c (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag ⋙ proj X L ⋙ L)))
  (IsLimit.lift (hc (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag ⋙ proj X L ⋙ L))
      { pt := X,
        π :=
          { app := fun (j : J) ↦ (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag.obj j).hom,
            naturality := sorry } } :
    Cone.pt.{v_3, v_1, u_3, u_1} (J := J) (F :=
        d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag ⋙ proj X L ⋙ L)
        { pt := X,
          π :=
            {
              app := fun (j : J) ↦
                (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag.obj j).hom,
              naturality := sorry :
              NatTrans (Functor.obj (Functor.const J) X)
                (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag ⋙ proj X L ⋙ L) } :
          Cone (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag ⋙ proj X L ⋙ L) } ⟶
      Cone.pt (F := d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag ⋙ proj X L ⋙ L)
        (c (d.toLimitPresentation (P := MorphismProperty.structuredArrowObj L P).diag ⋙ proj X L ⋙ L)))
Tactic: rw [P.structuredArrowObj_iff, heq, ← (hc _).lift_comp_conePointUniqueUpToIso_hom hd,
      P.cancel_right_of_respectsIso]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
X : T
inst_1 : HasPushouts.{v_1, u_1} T
inst_2 : P.IsStableUnderComposition
inst_3 : P.IsStableUnderCobaseChange
inst_4 : P.HasOfPrecompProperty P
⊢ ObjectProperty.IsClosedUnderColimitsOfShape (C := Under.{v_1, u_1} X) P.underObj WalkingSpan
have rw : ObjectProperty.IsClosedUnderLimitsOfShape (C := Over.{v_1, u_1} (Opposite.op X)) P.op.overObj WalkingCospan
Tactic: rw [ObjectProperty.isClosedUnderColimitsOfShape_iff_op, ←
    ObjectProperty.isClosedUnderLimitsOfShape_inverseImage_iff _ _ (Over.opEquivOpUnder _),
    MorphismProperty.inverseImage_op_underObj,
    ObjectProperty.isClosedUnderLimitsOfShape_iff_of_equivalence _ walkingSpanOpEquiv]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
X : T
inst_1 : P.ContainsIdentities
inst_2 : P.RespectsIso
⊢ CreatesLimitsOfShape (C := P.Over ⊤ X) (Discrete PEmpty.{1}) (Over.forget P ⊤ X)
have inst : HasLimitsOfShape (Discrete PEmpty.{1}) (Comma.{v_1, 0, v_1, u_1, 0, u_1} (𝟭 T) (Functor.fromPUnit X))
Tactic: apply +allowSynthFailures forgetCreatesLimitsOfShapeOfClosed
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst_1 : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
X : T
inst_2 : P.ContainsIdentities
inst_3 : P.RespectsIso
inst : HasLimitsOfShape (Discrete PEmpty.{1}) (Comma.{v_1, 0, v_1, u_1, 0, u_1} (𝟭 T) (Functor.fromPUnit X))
⊢ CreatesLimitsOfShape (C := P.Over ⊤ X) (Discrete PEmpty.{1}) (Over.forget P ⊤ X)
have inst₁ : ObjectProperty.IsClosedUnderLimitsOfShape (C := Comma.{v_1, 0, v_1, u_1, 0, u_1} (𝟭 T) (Functor.fromPUnit X))
  (commaObj (𝟭 T) (Functor.fromPUnit X) P) (Discrete PEmpty.{1})
Tactic: apply +allowSynthFailures forgetCreatesLimitsOfShapeOfClosed
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
X : T
inst_1 : P.ContainsIdentities
Y : P.Over ⊤ X
a : Y ⟶ Over.mk (P := P) ⊤ (𝟙 X) ⋯
⊢ a = Over.homMk (P := P) (Q := ⊤) Y.toComma (P := P) (Q := ⊤) (W := ⊤).hom ⋯ ⋯
have h : a.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
  CommaMorphism.left
    (Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
      (Over.homMk (P := P) (Q := ⊤) Y.toComma (P := P) (Q := ⊤) (W := ⊤).hom sorry sorry))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
X : T
inst_1 : P.ContainsIdentities
Y : P.Over ⊤ X
a : Y ⟶ Over.mk (P := P) ⊤ (𝟙 X) ⋯
⊢ a.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
    CommaMorphism.left
      (Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (Over.homMk (P := P) (Q := ⊤) Y.toComma (P := P) (Q := ⊤) (W := ⊤).hom ⋯ ⋯))
have h : a.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left = Y.toComma (P := P) (Q := ⊤) (W := ⊤).hom
Tactic: simp only [mk_left, homMk_hom, Over.homMk_left]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
X : T
inst_1 : P.ContainsIdentities
Y : P.Over ⊤ X
a : Y ⟶ Over.mk (P := P) ⊤ (𝟙 X) ⋯
⊢ a.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left = Y.toComma (P := P) (Q := ⊤) (W := ⊤).hom
have h : a.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
  a.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
    (toComma (P := P) (Q := ⊤) (W := ⊤) (Over.mk (P := P) ⊤ (𝟙 X) sorry)).hom
Tactic: rw [← Over.w a]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (hypothesis)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
X_1 : T
inst_1 : HasPullbacks.{v_1, u_1} T
inst_2 : P.IsStableUnderComposition
inst_3 : P.ContainsIdentities
inst_4 : P.IsStableUnderBaseChange
inst_5 : P.HasOfPostcompProperty P
X Y : T
f : X ⟶ Y
J : Type
x : SmallCategory J
x_1 : FinCategory J
⊢ PreservesLimitsOfShape (C := P.Over ⊤ Y) J (pullback P ⊤ f)
have this : PreservesLimitsOfShape (C := P.Over ⊤ Y) J (pullback P ⊤ f ⋙ Over.forget P ⊤ X)
Tactic: have : PreservesLimitsOfShape J
        (MorphismProperty.Over.pullback P ⊤ f ⋙ MorphismProperty.Over.forget _ _ _) :=
      inferInstanceAs <| PreservesLimitsOfShape J <|
        Over.forget _ _ _ ⋙ CategoryTheory.Over.pullback f
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
X : T
inst_1 : P.ContainsIdentities
inst_2 : P.RespectsIso
⊢ CreatesColimitsOfShape (C := P.Under ⊤ X) (Discrete PEmpty.{1}) (Under.forget P ⊤ X)
have inst : HasColimitsOfShape (Discrete PEmpty.{1}) (Comma.{0, v_1, v_1, 0, u_1, u_1} (Functor.fromPUnit X) (𝟭 T))
Tactic: apply +allowSynthFailures forgetCreatesColimitsOfShapeOfClosed
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.MorphismProperty (goal)
Goal:
T : Type u_1
inst_1 : Category.{v_1, u_1} T
P : MorphismProperty.{v_1, u_1} T
X : T
inst_2 : P.ContainsIdentities
inst_3 : P.RespectsIso
inst : HasColimitsOfShape (Discrete PEmpty.{1}) (Comma.{0, v_1, v_1, 0, u_1, u_1} (Functor.fromPUnit X) (𝟭 T))
⊢ CreatesColimitsOfShape (C := P.Under ⊤ X) (Discrete PEmpty.{1}) (Under.forget P ⊤ X)
have inst₁ : ObjectProperty.IsClosedUnderColimitsOfShape (C := Comma.{0, v_1, v_1, 0, u_1, u_1} (Functor.fromPUnit X) (𝟭 T))
  (commaObj (Functor.fromPUnit X) (𝟭 T) P) (Discrete PEmpty.{1})
Tactic: apply +allowSynthFailures forgetCreatesColimitsOfShapeOfClosed