awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
sC : C ≌ AsSmall C := AsSmall.equiv
F' : AsSmall C ⥤ Cat := sC.inverse ⋙ F ⋙ Cat.asSmallFunctor
G' : AsSmall C ⥤ Cat := sC.inverse ⋙ G ⋙ Cat.asSmallFunctor
α' : F' ⟶ G' := sC.inverse.whiskerLeft (whiskerRight α Cat.asSmallFunctor)
this : ∀ (X : AsSmall C), (α'.app X).toFunctor.Final
hα' : (map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor))).Final
⊢ (map α).Final
have have : ∀
  (i :
    map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor)) ≅
      (preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
        ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
          (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse),
  (map α).Final
Tactic: have i := mapWhiskerLeftIsoConjPreMap sC.symm (whiskerRight α Cat.asSmallFunctor)
    ≪≫ isoWhiskerLeft _ (isoWhiskerRight (mapWhiskerRightAsSmallFunctor α) _)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
sC : C ≌ AsSmall C := AsSmall.equiv
F' : AsSmall C ⥤ Cat := sC.inverse ⋙ F ⋙ Cat.asSmallFunctor
G' : AsSmall C ⥤ Cat := sC.inverse ⋙ G ⋙ Cat.asSmallFunctor
α' : F' ⟶ G' := sC.inverse.whiskerLeft (whiskerRight α Cat.asSmallFunctor)
this : ∀ (X : AsSmall C), (α'.app X).toFunctor.Final
hα' : (map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor))).Final
i :
  map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor)) ≅
    (preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
      ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse
⊢ (map α).Final
have this₁ : ((preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
    ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
      (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse).Final
Tactic: have := final_of_natIso i
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
sC : C ≌ AsSmall C := AsSmall.equiv
F' : AsSmall C ⥤ Cat := sC.inverse ⋙ F ⋙ Cat.asSmallFunctor
G' : AsSmall C ⥤ Cat := sC.inverse ⋙ G ⋙ Cat.asSmallFunctor
α' : F' ⟶ G' := sC.inverse.whiskerLeft (whiskerRight α Cat.asSmallFunctor)
this_1 : ∀ (X : AsSmall C), (α'.app X).toFunctor.Final
hα' : (map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor))).Final
i :
  map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor)) ≅
    (preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
      ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse
this :
  ((preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
      ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse).Final
⊢ (map α).Final
have rwa : ∀
  (this :
    (((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse).Final),
  (map α).Final
Tactic: rwa [← final_iff_equivalence_comp, ← final_iff_comp_equivalence,
    ← final_iff_equivalence_comp, ← final_iff_comp_equivalence] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
sC : C ≌ AsSmall C := AsSmall.equiv
F' : AsSmall C ⥤ Cat := sC.inverse ⋙ F ⋙ Cat.asSmallFunctor
G' : AsSmall C ⥤ Cat := sC.inverse ⋙ G ⋙ Cat.asSmallFunctor
α' : F' ⟶ G' := sC.inverse.whiskerLeft (whiskerRight α Cat.asSmallFunctor)
this_1 : ∀ (X : AsSmall C), (α'.app X).toFunctor.Final
hα' : (map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor))).Final
i :
  map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor)) ≅
    (preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
      ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse
this :
  ((preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
      ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse).Final
rwa :
  ∀
    (this :
      (((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
          (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse).Final),
    (map α).Final
⊢ (map α).Final
have rwa₁ : ∀ (this : ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse).Final),
  (map α).Final
Tactic: rwa [← final_iff_equivalence_comp, ← final_iff_comp_equivalence,
    ← final_iff_equivalence_comp, ← final_iff_comp_equivalence] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
sC : C ≌ AsSmall C := AsSmall.equiv
F' : AsSmall C ⥤ Cat := sC.inverse ⋙ F ⋙ Cat.asSmallFunctor
G' : AsSmall C ⥤ Cat := sC.inverse ⋙ G ⋙ Cat.asSmallFunctor
α' : F' ⟶ G' := sC.inverse.whiskerLeft (whiskerRight α Cat.asSmallFunctor)
this_1 : ∀ (X : AsSmall C), (α'.app X).toFunctor.Final
hα' : (map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor))).Final
i :
  map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor)) ≅
    (preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
      ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse
this :
  ((preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
      ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse).Final
rwa :
  ∀
    (this :
      (((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
          (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse).Final),
    (map α).Final
rwa₁ :
  ∀ (this : ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse).Final),
    (map α).Final
⊢ (map α).Final
have rwa₂ : ∀ (this : (map α ⋙ (compAsSmallFunctorEquivalence G).inverse).Final), (map α).Final
Tactic: rwa [← final_iff_equivalence_comp, ← final_iff_comp_equivalence,
    ← final_iff_equivalence_comp, ← final_iff_comp_equivalence] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
sC : C ≌ AsSmall C := AsSmall.equiv
F' : AsSmall C ⥤ Cat := sC.inverse ⋙ F ⋙ Cat.asSmallFunctor
G' : AsSmall C ⥤ Cat := sC.inverse ⋙ G ⋙ Cat.asSmallFunctor
α' : F' ⟶ G' := sC.inverse.whiskerLeft (whiskerRight α Cat.asSmallFunctor)
this_1 : ∀ (X : AsSmall C), (α'.app X).toFunctor.Final
hα' : (map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor))).Final
i :
  map (sC.symm.functor.whiskerLeft (whiskerRight α Cat.asSmallFunctor)) ≅
    (preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
      ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse
this :
  ((preEquivalence (F ⋙ Cat.asSmallFunctor) sC.symm).functor ⋙
      ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
        (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse).Final
rwa :
  ∀
    (this :
      (((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse) ⋙
          (preEquivalence (G ⋙ Cat.asSmallFunctor) sC.symm).inverse).Final),
    (map α).Final
rwa₁ :
  ∀ (this : ((compAsSmallFunctorEquivalence F).functor ⋙ map α ⋙ (compAsSmallFunctorEquivalence G).inverse).Final),
    (map α).Final
rwa₂ : ∀ (this : (map α ⋙ (compAsSmallFunctorEquivalence G).inverse).Final), (map α).Final
⊢ (map α).Final
have rwa₃ : ∀ (this : (map α).Final), (map α).Final
Tactic: rwa [← final_iff_equivalence_comp, ← final_iff_comp_equivalence,
    ← final_iff_equivalence_comp, ← final_iff_comp_equivalence] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : ObjectProperty.{v₁, u₁} C
h : ∀ (d : C), ¬P d → IsConnected (CostructuredArrow P.ι d)
d : C
⊢ IsConnected (CostructuredArrow P.ι d)
have pos : ∀ (hd : P d), IsConnected (CostructuredArrow P.ι d)
Tactic: by_cases hd : P d
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : ObjectProperty.{v₁, u₁} C
h : ∀ (d : C), ¬P d → IsConnected (CostructuredArrow P.ι d)
d : C
pos : ∀ (hd : P d), IsConnected (CostructuredArrow P.ι d)
⊢ IsConnected (CostructuredArrow P.ι d)
have neg : ∀ (hd : ¬P d), IsConnected (CostructuredArrow P.ι d)
Tactic: by_cases hd : P d
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : ObjectProperty.{v₁, u₁} C
h : ∀ (d : C), ¬P d → IsConnected (CostructuredArrow P.ι d)
d : C
hd : P d
⊢ IsConnected (CostructuredArrow P.ι d)
have this : Nonempty (CostructuredArrow P.ι d)
Tactic: have : Nonempty (CostructuredArrow P.ι d) := ⟨⟨d, hd⟩, ⟨⟨⟩⟩, 𝟙 _⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : ObjectProperty.{v₁, u₁} C
h : ∀ (d : C), ¬P d → IsConnected (CostructuredArrow P.ι d)
d : C
hd : P d
this : Nonempty (CostructuredArrow P.ι d)
⊢ IsConnected (CostructuredArrow P.ι d)
have pos : (j₁ j₂ : CostructuredArrow P.ι d) → j₁ ⟶ CostructuredArrow.mk (𝟙 (P.ι.obj (FullSubcategory.mk (P := P) d hd)))
Tactic: refine zigzag_isConnected (fun j₁ j₂ ↦ Zigzag.trans
      (j₂ := by exact CostructuredArrow.mk (Y := ⟨d, hd⟩) (𝟙 _)) (.of_hom ?_) (.of_inv ?_))
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : ObjectProperty.{v₁, u₁} C
h : ∀ (d : C), ¬P d → IsConnected (CostructuredArrow P.ι d)
d : C
hd : P d
this : Nonempty (CostructuredArrow P.ι d)
pos : (j₁ j₂ : CostructuredArrow P.ι d) → j₁ ⟶ CostructuredArrow.mk (𝟙 (P.ι.obj (FullSubcategory.mk (P := P) d hd)))
⊢ IsConnected (CostructuredArrow P.ι d)
have pos₁ : (j₁ j₂ : CostructuredArrow P.ι d) → j₂ ⟶ CostructuredArrow.mk (𝟙 (P.ι.obj (FullSubcategory.mk (P := P) d hd)))
Tactic: refine zigzag_isConnected (fun j₁ j₂ ↦ Zigzag.trans
      (j₂ := by exact CostructuredArrow.mk (Y := ⟨d, hd⟩) (𝟙 _)) (.of_hom ?_) (.of_inv ?_))
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (hypothesis)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
D : Functor.{v_1, v_2, u_1, u_2} J C
c : Cone D
hc : IsLimit c
j : J
inst_2 : Functor.Initial (C := Over.{v_1, u_1} j) (Over.forget j)
⊢ IsLimit (c.overPost j)
have this : Nonempty (Over.{v_1, u_1} j)
Tactic: haveI : Nonempty (Over j) := ⟨Over.mk (𝟙 j)⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
D : Functor.{v_1, v_2, u_1, u_2} J C
c : Cone D
hc : IsLimit c
j : J
inst_2 : Functor.Initial (C := Over.{v_1, u_1} j) (Over.forget j)
this : Nonempty (Over.{v_1, u_1} j)
c'' : Cone
  (Over.lift (Over.forget j ⋙ D)
    (Functor.whiskerRight (Cocone.ι (Over.forgetCocone j)) D ≫
      Iso.hom (Functor.constComp (Over.{v_1, u_1} j) (Cocone.pt (J := Over.{v_1, u_1} j) (Over.forgetCocone j)) D))) :=
  Over.liftCone (Over.forget j ⋙ D)
    (Functor.whiskerRight (Cocone.ι (Over.forgetCocone j)) D ≫
      Iso.hom (Functor.constComp (Over.{v_1, u_1} j) (Cocone.pt (J := Over.{v_1, u_1} j) (Over.forgetCocone j)) D))
    (Cone.whisker (Over.forget j) c) (c.π.app j) ⋯
hc'' : IsLimit c'' :=
  Over.isLimitLiftCone (Over.forget j ⋙ D)
    (Functor.whiskerRight (Cocone.ι (Over.forgetCocone j)) D ≫
      Iso.hom (Functor.constComp (Over.{v_1, u_1} j) (Cocone.pt (J := Over.{v_1, u_1} j) (Over.forgetCocone j)) D))
    (Cone.whisker (Over.forget j) c) (c.π.app j) ⋯
    ((Equiv.symm (Functor.Initial.isLimitWhiskerEquiv (Over.forget j) c) :
        (a : IsLimit c) → IsLimit (Cone.whisker (Over.forget j) c))
      hc)
⊢ IsLimit (c.overPost j)
have refine_1 : Over.lift (Over.forget j ⋙ D)
    (Functor.whiskerRight (Cocone.ι (Over.forgetCocone j)) D ≫
      Iso.hom (Functor.constComp (Over.{v_1, u_1} j) (Cocone.pt (J := Over.{v_1, u_1} j) (Over.forgetCocone j)) D)) ≅
  Over.post D
Tactic: refine IsLimit.equivOfNatIsoOfIso ?_ _ _ ?_ hc''
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
D : Functor.{v_1, v_2, u_1, u_2} J C
c : Cone D
hc : IsLimit c
j : J
inst_2 : Functor.Initial (C := Over.{v_1, u_1} j) (Over.forget j)
this : Nonempty (Over.{v_1, u_1} j)
c'' : Cone
  (Over.lift (Over.forget j ⋙ D)
    (Functor.whiskerRight (Cocone.ι (Over.forgetCocone j)) D ≫
      Iso.hom (Functor.constComp (Over.{v_1, u_1} j) (Cocone.pt (J := Over.{v_1, u_1} j) (Over.forgetCocone j)) D))) :=
  Over.liftCone (Over.forget j ⋙ D)
    (Functor.whiskerRight (Cocone.ι (Over.forgetCocone j)) D ≫
      Iso.hom (Functor.constComp (Over.{v_1, u_1} j) (Cocone.pt (J := Over.{v_1, u_1} j) (Over.forgetCocone j)) D))
    (Cone.whisker (Over.forget j) c) (c.π.app j) ⋯
hc'' : IsLimit c'' :=
  Over.isLimitLiftCone (Over.forget j ⋙ D)
    (Functor.whiskerRight (Cocone.ι (Over.forgetCocone j)) D ≫
      Iso.hom (Functor.constComp (Over.{v_1, u_1} j) (Cocone.pt (J := Over.{v_1, u_1} j) (Over.forgetCocone j)) D))
    (Cone.whisker (Over.forget j) c) (c.π.app j) ⋯
    ((Equiv.symm (Functor.Initial.isLimitWhiskerEquiv (Over.forget j) c) :
        (a : IsLimit c) → IsLimit (Cone.whisker (Over.forget j) c))
      hc)
refine_1 :
  Over.lift (Over.forget j ⋙ D)
      (Functor.whiskerRight (Cocone.ι (Over.forgetCocone j)) D ≫
        Iso.hom (Functor.constComp (Over.{v_1, u_1} j) (Cocone.pt (J := Over.{v_1, u_1} j) (Over.forgetCocone j)) D)) ≅
    Over.post D
⊢ IsLimit (c.overPost j)
have refine_2 : Functor.obj
    (Cone.postcompose
      (Iso.hom
        (NatIso.ofComponents
          (fun (k : Over.{v_1, u_1} j) ↦
            Over.isoMk
              (Iso.refl
                ((Over.lift (Over.forget j ⋙ D)
                        (Functor.whiskerRight (Cocone.ι (Over.forgetCocone j)) D ≫
                          Iso.hom
                            (Functor.constComp (Over.{v_1, u_1} j)
                              (Cocone.pt (J := Over.{v_1, u_1} j) (Over.forgetCocone j)) D))).obj
                    k).left)
              sorry)
          sorry)))
    c'' ≅
  c.overPost j
Tactic: refine IsLimit.equivOfNatIsoOfIso ?_ _ _ ?_ hc''
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (hypothesis)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
D : Functor.{v_1, v_2, u_1, u_2} J C
c : Cocone D
hc : IsColimit c
j : J
inst_2 : Functor.Final (C := Under.{v_1, u_1} j) (Under.forget j)
⊢ IsColimit (c.underPost j)
have this : Nonempty (Under.{v_1, u_1} j)
Tactic: haveI : Nonempty (Under j) := ⟨CategoryTheory.Under.mk (𝟙 j)⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
D : Functor.{v_1, v_2, u_1, u_2} J C
c : Cocone D
hc : IsColimit c
j : J
inst_2 : Functor.Final (C := Under.{v_1, u_1} j) (Under.forget j)
this : Nonempty (Under.{v_1, u_1} j)
c'' : Cocone
  (Under.lift (Under.forget j ⋙ D)
    (Iso.inv (Functor.constComp (Under.{v_1, u_1} j) j D) ≫ Functor.whiskerRight (Cone.π (Under.forgetCone j)) D)) :=
  Under.liftCocone (Under.forget j ⋙ D)
    (Iso.inv (Functor.constComp (Under.{v_1, u_1} j) j D) ≫ Functor.whiskerRight (Cone.π (Under.forgetCone j)) D)
    (Cocone.whisker (Under.forget j) c) (c.ι.app j) ⋯
hc'' : IsColimit c'' :=
  Under.isColimitLiftCocone (Under.forget j ⋙ D)
    (Iso.inv (Functor.constComp (Under.{v_1, u_1} j) j D) ≫ Functor.whiskerRight (Cone.π (Under.forgetCone j)) D)
    (Cocone.whisker (Under.forget j) c) (c.ι.app j) ⋯
    ((Equiv.symm (Functor.Final.isColimitWhiskerEquiv (Under.forget j) c) :
        (a : IsColimit c) → IsColimit (Cocone.whisker (Under.forget j) c))
      hc)
⊢ IsColimit (c.underPost j)
have refine_1 : Under.lift (Under.forget j ⋙ D)
    (Iso.inv (Functor.constComp (Under.{v_1, u_1} j) j D) ≫ Functor.whiskerRight (Cone.π (Under.forgetCone j)) D) ≅
  Under.post D
Tactic: refine IsColimit.equivOfNatIsoOfIso ?_ _ _ ?_ hc''
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
D : Functor.{v_1, v_2, u_1, u_2} J C
c : Cocone D
hc : IsColimit c
j : J
inst_2 : Functor.Final (C := Under.{v_1, u_1} j) (Under.forget j)
this : Nonempty (Under.{v_1, u_1} j)
c'' : Cocone
  (Under.lift (Under.forget j ⋙ D)
    (Iso.inv (Functor.constComp (Under.{v_1, u_1} j) j D) ≫ Functor.whiskerRight (Cone.π (Under.forgetCone j)) D)) :=
  Under.liftCocone (Under.forget j ⋙ D)
    (Iso.inv (Functor.constComp (Under.{v_1, u_1} j) j D) ≫ Functor.whiskerRight (Cone.π (Under.forgetCone j)) D)
    (Cocone.whisker (Under.forget j) c) (c.ι.app j) ⋯
hc'' : IsColimit c'' :=
  Under.isColimitLiftCocone (Under.forget j ⋙ D)
    (Iso.inv (Functor.constComp (Under.{v_1, u_1} j) j D) ≫ Functor.whiskerRight (Cone.π (Under.forgetCone j)) D)
    (Cocone.whisker (Under.forget j) c) (c.ι.app j) ⋯
    ((Equiv.symm (Functor.Final.isColimitWhiskerEquiv (Under.forget j) c) :
        (a : IsColimit c) → IsColimit (Cocone.whisker (Under.forget j) c))
      hc)
refine_1 :
  Under.lift (Under.forget j ⋙ D)
      (Iso.inv (Functor.constComp (Under.{v_1, u_1} j) j D) ≫ Functor.whiskerRight (Cone.π (Under.forgetCone j)) D) ≅
    Under.post D
⊢ IsColimit (c.underPost j)
have refine_2 : Functor.obj
    (Cocone.precompose
      (Iso.inv
        (NatIso.ofComponents
          (fun (k : Under.{v_1, u_1} j) ↦
            Under.isoMk
              (Iso.refl
                ((Under.lift (Under.forget j ⋙ D)
                        (Iso.inv (Functor.constComp (Under.{v_1, u_1} j) j D) ≫
                          Functor.whiskerRight (Cone.π (Under.forgetCone j)) D)).obj
                    k).right)
              sorry)
          sorry)))
    c'' ≅
  c.underPost j
Tactic: refine IsColimit.equivOfNatIsoOfIso ?_ _ _ ?_ hc''
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C₀ : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} C₀
inst_1 : Category.{v_2, u_2} C
F : Functor.{v_1, v_2, u_1, u_2} C₀ C
X : C
inst_2 : F.Initial
Y : Over.{v_2, u_2} X
⊢ IsConnected (CostructuredArrow (CostructuredArrow.toOver F X) Y)
have rw : IsConnected (CostructuredArrow F Y.left)
Tactic: rw [isConnected_iff_of_equivalence
      (CostructuredArrow.costructuredArrowToOverEquivalence F Y)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.FinallySmall (hypothesis)
Goal:
J : Type u
inst : Category.{v, u} J
K : Type u₁
inst_1 : Category.{v₁, u₁} K
inst_2 : LocallySmall.{w, v, u} J
inst_3 : InitiallySmall.{w, v, u} J
X : J
⊢ InitiallySmall (Over.{v, u} X)
have this : InitiallySmall (CostructuredArrow.{w, v, w, u} (C := InitialModel.{w, v, u} J) (fromInitialModel J) X)
Tactic: have : InitiallySmall.{w} (CostructuredArrow (fromInitialModel.{w} J) X) :=
    initiallySmall_of_essentiallySmall _
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.FinallySmall (goal)
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : FinallySmall.{w, v, u} J
⊢ ∃ (s : Set J) (_ : Small.{w, u} (↑s : Type u)), ∀ (i : J), ∃ j ∈ s, Nonempty.{v + 1} (i ⟶ j)
have refine : ∀ (i : J), ∃ j ∈ Set.range.{u, w + 1} (ι := FinalModel.{w, v, u} J) (obj (fromFinalModel J)), Nonempty.{v + 1} (i ⟶ j)
Tactic: refine ⟨Set.range (fromFinalModel J).obj, inferInstance, fun i => ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.FinallySmall (goal)
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : FinallySmall.{w, v, u} J
i : J
⊢ ∃ j ∈ Set.range.{u, w + 1} (ι := FinalModel.{w, v, u} J) (obj (fromFinalModel J)), Nonempty.{v + 1} (i ⟶ j)
have obtain : ∀ (f : StructuredArrow.{w, v, w, u} (C := FinalModel.{w, v, u} J) i (fromFinalModel J)),
  ∃ j ∈ Set.range.{u, w + 1} (ι := FinalModel.{w, v, u} J) (obj (fromFinalModel J)), Nonempty.{v + 1} (i ⟶ j)
Tactic: obtain ⟨f⟩ : Nonempty (StructuredArrow i (fromFinalModel J)) := IsConnected.is_nonempty
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.FinallySmall (goal)
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : IsFilteredOrEmpty.{v, u} J
s : Set J
inst_2 : Small.{v, u} (↑s : Type u)
hs : ∀ (i : J), ∃ j ∈ s, Nonempty.{v + 1} (i ⟶ j)
⊢ FinallySmall.{v, v, u} J
have suffices : Final.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (C := J) fun (x : J) ↦ x ∈ s)
  (ObjectProperty.ι fun (x : J) ↦ x ∈ s)
Tactic: suffices Functor.Final (ObjectProperty.ι (· ∈ s)) from
    finallySmall_of_final_of_essentiallySmall (ObjectProperty.ι (· ∈ s))
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.FinallySmall (goal)
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : IsFilteredOrEmpty.{v, u} J
s : Set J
inst_2 : Small.{v, u} (↑s : Type u)
hs : ∀ (i : J), ∃ j ∈ s, Nonempty.{v + 1} (i ⟶ j)
⊢ Final.{v, v, u, u} (C := ObjectProperty.FullSubcategory.{v, u} (C := J) fun (x : J) ↦ x ∈ s)
    (ObjectProperty.ι fun (x : J) ↦ x ∈ s)
have refine : ∀ (i : J),
  ∃ (c : ObjectProperty.FullSubcategory.{v, u} (C := J) fun (x : J) ↦ x ∈ s),
    Nonempty.{v + 1} (i ⟶ obj (ObjectProperty.ι fun (x : J) ↦ x ∈ s) c)
Tactic: refine Functor.final_of_exists_of_isFiltered_of_fullyFaithful _ (fun i => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.FinallySmall (goal)
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : IsFilteredOrEmpty.{v, u} J
s : Set J
inst_2 : Small.{v, u} (↑s : Type u)
hs : ∀ (i : J), ∃ j ∈ s, Nonempty.{v + 1} (i ⟶ j)
i : J
⊢ ∃ (c : ObjectProperty.FullSubcategory.{v, u} (C := J) fun (x : J) ↦ x ∈ s),
    Nonempty.{v + 1} (i ⟶ obj (ObjectProperty.ι fun (x : J) ↦ x ∈ s) c)
have obtain : ∀ (j : J) (hj₁ : j ∈ s) (hj₂ : Nonempty.{v + 1} (i ⟶ j)),
  ∃ (c : ObjectProperty.FullSubcategory.{v, u} (C := J) fun (x : J) ↦ x ∈ s),
    Nonempty.{v + 1} (i ⟶ obj (ObjectProperty.ι fun (x : J) ↦ x ∈ s) c)
Tactic: obtain ⟨j, hj₁, hj₂⟩ := hs i
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.FinallySmall (goal)
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : IsFilteredOrEmpty.{v, u} J
⊢ FinallySmall.{v, v, u} J ↔
    ∃ (s : Set J) (_ : Small.{v, u} (↑s : Type u)), ∀ (i : J), ∃ j ∈ s, Nonempty.{v + 1} (i ⟶ j)
have refine : ∀ (h : ∃ (s : Set J) (_ : Small.{v, u} (↑s : Type u)), ∀ (i : J), ∃ j ∈ s, Nonempty.{v + 1} (i ⟶ j)),
  FinallySmall.{v, v, u} J
Tactic: refine ⟨fun _ => FinallySmall.exists_small_weakly_terminal_set _, fun h => ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.FinallySmall (goal)
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : IsFilteredOrEmpty.{v, u} J
h : ∃ (s : Set J) (_ : Small.{v, u} (↑s : Type u)), ∀ (i : J), ∃ j ∈ s, Nonempty.{v + 1} (i ⟶ j)
⊢ FinallySmall.{v, v, u} J
have rcases : ∀ (s : Set J) (hs : Small.{v, u} (↑s : Type u)) (hs' : ∀ (i : J), ∃ j ∈ s, Nonempty.{v + 1} (i ⟶ j)),
  FinallySmall.{v, v, u} J
Tactic: rcases h with ⟨s, hs, hs'⟩
Cleared hypotheses: h