awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : IsFiltered.{v, u} J
F : Functor.{v, w₀, u, w₀ + 1} J (Type w₀)
c : CoconeTypes.{w₁, w₀, v, u} F
h :
  ∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),
    c.ι j x = c.ι j' x' →
      ∃ (k : J) (f : j ⟶ k) (f' : j' ⟶ k),
        Eq (α := (fun (X : Type w₀) ↦ X) (F.obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
              x :
            (fun (X : Type w₀) ↦ X) (F.obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f') :
                (a : (fun (X : Type w₀) ↦ X) (F.obj j')) → (fun (X : Type w₀) ↦ X) (F.obj k))
              x' :
            (fun (X : Type w₀) ↦ X) (F.obj k))
j : J
x x' : F.obj j
eq_1 : c.ι j x = c.ι j x'
k : J
f f' : j ⟶ k
eq :
  Eq (α := (fun (X : Type w₀) ↦ X) (F.obj k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
          (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
        x :
      (fun (X : Type w₀) ↦ X) (F.obj k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f') :
          (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
        x' :
      (fun (X : Type w₀) ↦ X) (F.obj k))
mp mp₁ :
  Eq (α := F.obj (coeq f f'))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (f' ≫ coeqHom f f')) : (a : F.obj j) → F.obj (coeq f f')) x' :
      F.obj (coeq f f'))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (f' ≫ coeqHom f f')) : (a : F.obj j) → F.obj (coeq f f')) x' :
      F.obj (coeq f f'))
⊢ Eq (α := F.obj (coeq f f'))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (f' ≫ coeqHom f f')) : (a : F.obj j) → F.obj (coeq f f')) x' :
      F.obj (coeq f f'))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (f ≫ coeqHom f f')) : (a : F.obj j) → F.obj (coeq f f')) x' :
      F.obj (coeq f f'))

Tactic:
rw [coeq_condition]

Premises used: CategoryTheory.IsFiltered.coeq_condition
-- Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : IsFiltered.{v, u} J
F : Functor.{v, w₀, u, w₀ + 1} J (Type w₀)
c : CoconeTypes.{w₁, w₀, v, u} F
h :
  ∀ (j : J) (x x' : F.obj j),
    c.ι j x = c.ι j x' →
      ∃ (k : J) (f : j ⟶ k),
        Eq (α := (fun (X : Type w₀) ↦ X) (F.obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
              x :
            (fun (X : Type w₀) ↦ X) (F.obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
              x' :
            (fun (X : Type w₀) ↦ X) (F.obj k))
j j' : J
x : F.obj j
x' : F.obj j'
eq : c.ι j x = c.ι j' x'
⊢ c.ι (IsFiltered.max.{v, u} j j')
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (leftToMax j j')) :
          (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j')))
        x) =
    c.ι (IsFiltered.max.{v, u} j j')
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (rightToMax j j')) :
          (a : (fun (X : Type w₀) ↦ X) (F.obj j')) → (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j')))
        x')

Tactic:
simpa only [c.ι_naturality_apply]

Premises used: CategoryTheory.Functor.CoconeTypes.ι_naturality_apply
-- Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : IsFiltered.{v, u} J
F : Functor.{v, w₀, u, w₀ + 1} J (Type w₀)
c : CoconeTypes.{w₁, w₀, v, u} F
h :
  ∀ (j : J) (x x' : F.obj j),
    c.ι j x = c.ι j x' →
      ∃ (k : J) (f : j ⟶ k),
        Eq (α := (fun (X : Type w₀) ↦ X) (F.obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
              x :
            (fun (X : Type w₀) ↦ X) (F.obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
              x' :
            (fun (X : Type w₀) ↦ X) (F.obj k))
j j' : J
x : F.obj j
x' : F.obj j'
eq_1 : c.ι j x = c.ι j' x'
k : J
g : IsFiltered.max.{v, u} j j' ⟶ k
eq :
  Eq (α := (fun (X : Type w₀) ↦ X) (F.obj k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g) :
          (a : (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j'))) → (fun (X : Type w₀) ↦ X) (F.obj k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (leftToMax j j')) :
              (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j')))
            x :
          (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max.{v, u} j j'))) :
      (fun (X : Type w₀) ↦ X) (F.obj k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g) :
          (a : (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j'))) → (fun (X : Type w₀) ↦ X) (F.obj k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (rightToMax j j')) :
              (a : (fun (X : Type w₀) ↦ X) (F.obj j')) → (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j')))
            x' :
          (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max.{v, u} j j'))) :
      (fun (X : Type w₀) ↦ X) (F.obj k))
⊢ Eq (α := (fun (X : Type w₀) ↦ X) (F.obj k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (leftToMax j j' ≫ g)) :
          (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
        x :
      (fun (X : Type w₀) ↦ X) (F.obj k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (rightToMax j j' ≫ g)) :
          (a : (fun (X : Type w₀) ↦ X) (F.obj j')) → (fun (X : Type w₀) ↦ X) (F.obj k))
        x' :
      (fun (X : Type w₀) ↦ X) (F.obj k))

Tactic:
simp [eq]
-- Module: Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered
Goal:
J : Type u
inst : Category.{v, u} J
inst_1 : IsFiltered.{v, u} J
F : Functor.{v, w₀, u, w₀ + 1} J (Type w₀)
c : CoconeTypes.{w₁, w₀, v, u} F
h :
  ∀ (j : J) (x x' : F.obj j),
    c.ι j x = c.ι j x' →
      ∃ (k : J) (f : j ⟶ k),
        Eq (α := (fun (X : Type w₀) ↦ X) (F.obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
              x :
            (fun (X : Type w₀) ↦ X) (F.obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
              x' :
            (fun (X : Type w₀) ↦ X) (F.obj k))
j j' : J
x : F.obj j
x' : F.obj j'
eq_1 : c.ι j x = c.ι j' x'
k : J
g : IsFiltered.max.{v, u} j j' ⟶ k
eq :
  Eq (α := (fun (X : Type w₀) ↦ X) (F.obj k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g) :
          (a : (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j'))) → (fun (X : Type w₀) ↦ X) (F.obj k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (leftToMax j j')) :
              (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j')))
            x :
          (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max.{v, u} j j'))) :
      (fun (X : Type w₀) ↦ X) (F.obj k))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g) :
          (a : (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j'))) → (fun (X : Type w₀) ↦ X) (F.obj k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (rightToMax j j')) :
              (a : (fun (X : Type w₀) ↦ X) (F.obj j')) → (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max j j')))
            x' :
          (fun (X : Type w₀) ↦ X) (F.obj (IsFiltered.max.{v, u} j j'))) :
      (fun (X : Type w₀) ↦ X) (F.obj k))
⊢ ∃ (k : J) (f : j ⟶ k) (f' : j' ⟶ k),
    Eq (α := (fun (X : Type w₀) ↦ X) (F.obj k))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
            (a : (fun (X : Type w₀) ↦ X) (F.obj j)) → (fun (X : Type w₀) ↦ X) (F.obj k))
          x :
        (fun (X : Type w₀) ↦ X) (F.obj k))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f') :
            (a : (fun (X : Type w₀) ↦ X) (F.obj j')) → (fun (X : Type w₀) ↦ X) (F.obj k))
          x' :
        (fun (X : Type w₀) ↦ X) (F.obj k))

Tactic:
exact ⟨k, leftToMax _ _ ≫ g, rightToMax _ _ ≫ g, by simp [eq]⟩
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
inst_1 : Small.{u, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
⊢ Small.{u, max u v} F.ColimitType

Tactic:
small_of_surjective Quot.mk_surjective

Premises used: small_of_surjective, Quot.mk_surjective
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cocone F
j j' : J
f : j ⟶ j'
h :
  ∀ (x : (fun (X : Type u) ↦ X) (F.obj j)),
    Eq (α := c.pt)
      (Function.comp (β := F.obj j')
          (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j')) :
            (a : (fun (X : Type u) ↦ X) (F.obj j')) → (fun (X : Type u) ↦ X) (((const J).obj c.pt).obj j'))
          (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f)) :
            (a : (fun (X : Type u) ↦ X) (F.obj j)) → (fun (X : Type u) ↦ X) (F.obj j'))
          x :
        c.pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) :
            (a : (fun (X : Type u) ↦ X) (F.obj j)) → (fun (X : Type u) ↦ X) (((const J).obj c.pt).obj j))
          x :
        (fun (X : Type u) ↦ X) (obj.{w, u, v, u + 1} (obj (const J) c.pt) j))
⊢ Eq (α := F.obj j → c.pt)
    (Function.comp (β := F.obj j')
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j')) :
        (a : (fun (X : Type u) ↦ X) (F.obj j')) → (fun (X : Type u) ↦ X) (((const J).obj c.pt).obj j'))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f)) :
        (a : (fun (X : Type u) ↦ X) (F.obj j)) → (fun (X : Type u) ↦ X) (F.obj j')))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j)) :
      (a : (fun (X : Type u) ↦ X) (F.obj j)) → (fun (X : Type u) ↦ X) (((const J).obj c.pt).obj j))

Tactic:
ext x
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cocone F
j j' : J
f : j ⟶ j'
x : (fun (X : Type u) ↦ X) (F.obj j)
⊢ Eq (α := c.pt)
    (Function.comp (β := F.obj j')
        (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j')) :
          (a : (fun (X : Type u) ↦ X) (F.obj j')) → (fun (X : Type u) ↦ X) (((const J).obj c.pt).obj j'))
        (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f)) :
          (a : (fun (X : Type u) ↦ X) (F.obj j)) → (fun (X : Type u) ↦ X) (F.obj j'))
        x :
      c.pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) :
          (a : (fun (X : Type u) ↦ X) (F.obj j)) → (fun (X : Type u) ↦ X) (((const J).obj c.pt).obj j))
        x :
      (fun (X : Type u) ↦ X) (obj.{w, u, v, u + 1} (obj (const J) c.pt) j))

Tactic:
ConcreteCategory.congr_hom (c.w f) x

Premises used: CategoryTheory.Limits.Cocone.w, CategoryTheory.ConcreteCategory.congr_hom
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cocone F
j j' : J
f : j ⟶ j'
x : (fun (X : Type u) ↦ X) (F.obj j)
⊢ Eq (α := c.pt)
    (Function.comp (β := F.obj j')
        (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j')) :
          (a : (fun (X : Type u) ↦ X) (F.obj j')) → (fun (X : Type u) ↦ X) (((const J).obj c.pt).obj j'))
        (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f)) :
          (a : (fun (X : Type u) ↦ X) (F.obj j)) → (fun (X : Type u) ↦ X) (F.obj j'))
        x :
      c.pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) :
          (a : (fun (X : Type u) ↦ X) (F.obj j)) → (fun (X : Type u) ↦ X) (((const J).obj c.pt).obj j))
        x :
      (fun (X : Type u) ↦ X) (obj.{w, u, v, u + 1} (obj (const J) c.pt) j))

Tactic:
exact ConcreteCategory.congr_hom (c.w f) x

Premises used: CategoryTheory.Limits.Cocone.w, CategoryTheory.ConcreteCategory.congr_hom
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
mp :
  c.IsColimit →
    Nonempty
      (Limits.IsColimit.{w, u, v, u + 1} (J := J) (C := Type u) (F := F)
        ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c))
mpr :
  Nonempty
      (Limits.IsColimit.{w, u, v, u + 1} (J := J) (C := Type u) (F := F)
        ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) →
    c.IsColimit
⊢ c.IsColimit ↔
    Nonempty
      (Limits.IsColimit.{w, u, v, u + 1} (J := J) (C := Type u) (F := F)
        ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c))

Tactic:
constructor
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc : c.IsColimit
s : Cocone F
j : J
h :
  ∀ (x : F.obj j),
    TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryStruct.comp (Y :=
            obj.{w, u, v, u + 1}
              (obj (const J)
                (Cocone.pt.{w, u, v, u + 1} (J := J) (F := F)
                  ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)))
              j)
            (NatTrans.app (Cocone.ι ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) j)
            (↾fun
                (x :
                  Cocone.pt.{w, u, v, u + 1} (J := J) (F := F)
                    ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) ↦
              hc.desc ((F.coconeTypesEquiv.symm : (a : Cocone F) → F.CoconeTypes) s) x)))
        x =
      (ConcreteCategory.hom (FC := TypeCat.Fun) (s.ι.app j)).toFun x
⊢ CategoryStruct.comp (Y :=
      obj.{w, u, v, u + 1}
        (obj (const J)
          (Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)))
        j)
      (NatTrans.app (Cocone.ι ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) j)
      (↾fun
          (x : Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) ↦
        hc.desc ((F.coconeTypesEquiv.symm : (a : Cocone F) → F.CoconeTypes) s) x) =
    s.ι.app j

Tactic:
ext x
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc : c.IsColimit
s : Cocone F
j : J
x : F.obj j
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (CategoryStruct.comp (Y :=
          obj.{w, u, v, u + 1}
            (obj (const J)
              (Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)))
            j)
          (NatTrans.app (Cocone.ι ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) j)
          (↾fun
              (x :
                Cocone.pt.{w, u, v, u + 1} (J := J) (F := F)
                  ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) ↦
            hc.desc ((F.coconeTypesEquiv.symm : (a : Cocone F) → F.CoconeTypes) s) x)))
      x =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (s.ι.app j)).toFun x

Tactic:
congr_fun (hc.fac (F.coconeTypesEquiv.symm s) j) x

Premises used: CategoryTheory.Functor.coconeTypesEquiv, Classical.choose, congr_fun, Equiv.symm, CategoryTheory.Functor.CoconeTypes.IsColimit.desc, DFunLike.coe, CategoryTheory.Functor.CoconeTypes.IsColimit.fac
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc : c.IsColimit
s : Cocone F
j : J
x : F.obj j
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (CategoryStruct.comp (Y :=
          obj.{w, u, v, u + 1}
            (obj (const J)
              (Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)))
            j)
          (NatTrans.app (Cocone.ι ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) j)
          (↾fun
              (x :
                Cocone.pt.{w, u, v, u + 1} (J := J) (F := F)
                  ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) ↦
            hc.desc ((F.coconeTypesEquiv.symm : (a : Cocone F) → F.CoconeTypes) s) x)))
      x =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (s.ι.app j)).toFun x

Tactic:
exact congr_fun (hc.fac (F.coconeTypesEquiv.symm s) j) x

Premises used: CategoryTheory.Functor.coconeTypesEquiv, congr_fun, Equiv.symm, DFunLike.coe, CategoryTheory.Functor.CoconeTypes.IsColimit.fac
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc : c.IsColimit
s : Cocone F
m : Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c) ⟶ s.pt
hm : ∀ (j : J), NatTrans.app (Cocone.ι ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) j ≫ m = s.ι.app j
h :
  ∀ (x : Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)),
    (ConcreteCategory.hom (FC := TypeCat.Fun) m).toFun x =
      TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (↾fun
              (x :
                Cocone.pt.{w, u, v, u + 1} (J := J) (F := F)
                  ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) ↦
            hc.desc ((F.coconeTypesEquiv.symm : (a : Cocone F) → F.CoconeTypes) s) x))
        x
⊢ m =
    ↾fun (x : Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) ↦
      hc.desc ((F.coconeTypesEquiv.symm : (a : Cocone F) → F.CoconeTypes) s) x

Tactic:
ext x
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc : c.IsColimit
s : Cocone F
m : Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c) ⟶ s.pt
hm : ∀ (j : J), NatTrans.app (Cocone.ι ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) j ≫ m = s.ι.app j
x : Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)
j : J
y : F.obj j
⊢ ((ConcreteCategory.hom (FC := TypeCat.Fun) m).toFun ∘ c.ι j) y =
    (TypeCat.Fun.toFun
          (ConcreteCategory.hom (FC := TypeCat.Fun)
            (↾fun
                (x :
                  Cocone.pt.{w, u, v, u + 1} (J := J) (F := F)
                    ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) ↦
              hc.desc ((F.coconeTypesEquiv.symm : (a : Cocone F) → F.CoconeTypes) s) x)) ∘
        c.ι j)
      y

Tactic:
simp [← hm j]
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc : c.IsColimit
s : Cocone F
m : Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c) ⟶ s.pt
hm : ∀ (j : J), NatTrans.app (Cocone.ι ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) j ≫ m = s.ι.app j
x : Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) m).toFun x =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (↾fun
            (x :
              Cocone.pt.{w, u, v, u + 1} (J := J) (F := F) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)) ↦
          hc.desc ((F.coconeTypesEquiv.symm : (a : Cocone F) → F.CoconeTypes) s) x))
      x

Tactic:
exact congr_fun (hc.funext fun j ↦ funext fun y ↦ by simp [← hm j]) x

Premises used: funext, congr_fun, CategoryTheory.Functor.CoconeTypes.IsColimit.funext
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc : c.IsColimit
⊢ Nonempty
    (Limits.IsColimit.{w, u, v, u + 1} (J := J) (C := Type u) (F := F)
      ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c))

Tactic:
exact
     ⟨{ desc s := ↾fun x => hc.desc (F.coconeTypesEquiv.symm s) x
        fac s j := by
          ext x
          exact congr_fun (hc.fac (F.coconeTypesEquiv.symm s) j) x
        uniq s m hm := by
          ext x
          exact congr_fun (hc.funext fun j ↦ funext fun y ↦ by simp [← hm j]) x }⟩
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc :
  Limits.IsColimit.{w, u, v, u + 1} (J := J) (C := Type u) (F := F)
    ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)
⊢ c.IsColimit

Tactic:
⟨⟨fun x y h ↦ ?_, fun x ↦ ?_⟩⟩

Premises used: CategoryTheory.Functor.CoconeTypes.IsColimit.mk, And.intro
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc :
  Limits.IsColimit.{w, u, v, u + 1} (J := J) (C := Type u) (F := F)
    ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)
mpr : ∀ (x y : F.ColimitType) (h : F.descColimitType c x = F.descColimitType c y), x = y
mpr₁ : ∀ (x : c.pt), ∃ (a : F.ColimitType), F.descColimitType c a = x
⊢ c.IsColimit

Tactic:
refine ⟨⟨fun x y h ↦ ?_, fun x ↦ ?_⟩⟩
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc :
  Limits.IsColimit.{w, u, v, u + 1} (J := J) (C := Type u) (F := F)
    ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)
x y : F.ColimitType
h : F.descColimitType c x = F.descColimitType c y
f : (z : F.ColimitType) → ULift.{u, 0} Bool := fun (z : F.ColimitType) ↦ { down := decide (x = z) }
this : f x = f y
⊢ x = y

Tactic:
simpa [f] using this
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc :
  Limits.IsColimit.{w, u, v, u + 1} (J := J) (C := Type u) (F := F)
    ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)
x y : F.ColimitType
h : F.descColimitType c x = F.descColimitType c y
f : (z : F.ColimitType) → ULift.{u, 0} Bool := fun (z : F.ColimitType) ↦ { down := decide (x = z) }
mpr : f x = f y
⊢ x = y

Tactic:
suffices f x = f y by simpa [f] using this
-- Module: Mathlib.CategoryTheory.Limits.Types.Colimits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : F.CoconeTypes
hc :
  Limits.IsColimit.{w, u, v, u + 1} (J := J) (C := Type u) (F := F)
    ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c)
x y : F.ColimitType
h : F.descColimitType c x = F.descColimitType c y
f : (z : F.ColimitType) → ULift.{u, 0} Bool := fun (z : F.ColimitType) ↦ { down := decide (x = z) }
mpr :
  ∀ (z : F.ColimitType),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (hc.desc
              ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F)
                (F.coconeTypes.postcomp (T := ULift.{u, 0} Bool) f))) :
          (a : (fun (X : Type u) ↦ X) ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) c).pt) →
            (fun (X : Type u) ↦ X)
              ((F.coconeTypesEquiv : (a : F.CoconeTypes) → Cocone F) (F.coconeTypes.postcomp f)).pt)
        (F.descColimitType c z) =
      f z
⊢ f x = f y

Tactic:
suffices ∀ z, hc.desc (F.coconeTypesEquiv (F.coconeTypes.postcomp f))
          (F.descColimitType c z) = f z by rw [← this x, h, ← this y]