l3lab/ntp-mathlib huggingface dataset:
@[to_additive]
instance CommGroupCat.forget_reflects_isos : ReflectsIsomorphisms (forget CommGroupCat.{u}) where
  reflects {X Y} f _ := by
    let i := asIso ((forget CommGroupCat).map f)
    let e : X ≃* Y := { i.toEquiv with map_mul' := by aesop }
    

Proof state:
X Y : CommGroupCat
f : X ⟶ Y
x✝ : IsIso ((forget CommGroupCat).map f)
i : (forget CommGroupCat).obj X ≅ (forget CommGroupCat).obj Y := asIso ((forget CommGroupCat).map f)
e : ↑X ≃* ↑Y :=
  let src := i.toEquiv;
  {
    toEquiv :=
      { toFun := src.toFun, invFun := src.invFun, left_inv := (_ : Function.LeftInverse src.invFun src.toFun),
        right_inv := (_ : Function.RightInverse src.invFun src.toFun) },
    map_mul' := (_ : ∀ (x_1 y : ↑X), f (x_1 * y) = f x_1 * f y) }
⊢ IsIso f

Next tactic:
exact IsIso.of_iso e.toCommGroupCatIso
instance preservesLimit_subsingleton (K : J ⥤ C) (F : C ⥤ D) :
    Subsingleton (PreservesLimit K F) := by
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K✝ K : J ⥤ C
F : C ⥤ D
⊢ Subsingleton (PreservesLimit K F)

Next tactic:
constructor
instance preservesLimit_subsingleton (K : J ⥤ C) (F : C ⥤ D) :
    Subsingleton (PreservesLimit K F) := by
  constructor; 

Proof state:
case allEq
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K✝ K : J ⥤ C
F : C ⥤ D
⊢ ∀ (a b : PreservesLimit K F), a = b

Next tactic:
rintro ⟨a⟩ ⟨b⟩
instance preservesLimit_subsingleton (K : J ⥤ C) (F : C ⥤ D) :
    Subsingleton (PreservesLimit K F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; 

Proof state:
case allEq.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K✝ K : J ⥤ C
F : C ⥤ D
a b : {c : Cone K} → IsLimit c → IsLimit (F.mapCone c)
⊢ { preserves := a } = { preserves := b }

Next tactic:
congr!
instance preservesColimit_subsingleton (K : J ⥤ C) (F : C ⥤ D) :
    Subsingleton (PreservesColimit K F) := by
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K✝ K : J ⥤ C
F : C ⥤ D
⊢ Subsingleton (PreservesColimit K F)

Next tactic:
constructor
instance preservesColimit_subsingleton (K : J ⥤ C) (F : C ⥤ D) :
    Subsingleton (PreservesColimit K F) := by
  constructor; 

Proof state:
case allEq
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K✝ K : J ⥤ C
F : C ⥤ D
⊢ ∀ (a b : PreservesColimit K F), a = b

Next tactic:
rintro ⟨a⟩ ⟨b⟩
instance preservesColimit_subsingleton (K : J ⥤ C) (F : C ⥤ D) :
    Subsingleton (PreservesColimit K F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; 

Proof state:
case allEq.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K✝ K : J ⥤ C
F : C ⥤ D
a b : {c : Cocone K} → IsColimit c → IsColimit (F.mapCocone c)
⊢ { preserves := a } = { preserves := b }

Next tactic:
congr!
instance preservesLimitsOfShape_subsingleton (J : Type w) [Category.{w'} J] (F : C ⥤ D) :
    Subsingleton (PreservesLimitsOfShape J F) := by
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J✝ : Type w
inst✝¹ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
inst✝ : Category.{w', w} J
F : C ⥤ D
⊢ Subsingleton (PreservesLimitsOfShape J F)

Next tactic:
constructor
instance preservesLimitsOfShape_subsingleton (J : Type w) [Category.{w'} J] (F : C ⥤ D) :
    Subsingleton (PreservesLimitsOfShape J F) := by
  constructor; 

Proof state:
case allEq
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J✝ : Type w
inst✝¹ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
inst✝ : Category.{w', w} J
F : C ⥤ D
⊢ ∀ (a b : PreservesLimitsOfShape J F), a = b

Next tactic:
rintro ⟨a⟩ ⟨b⟩
instance preservesLimitsOfShape_subsingleton (J : Type w) [Category.{w'} J] (F : C ⥤ D) :
    Subsingleton (PreservesLimitsOfShape J F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; 

Proof state:
case allEq.mk.mk
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J✝ : Type w
inst✝¹ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
inst✝ : Category.{w', w} J
F : C ⥤ D
a b : {K : J ⥤ C} → PreservesLimit K F
⊢ PreservesLimitsOfShape.mk = PreservesLimitsOfShape.mk

Next tactic:
congr!
instance preservesColimitsOfShape_subsingleton (J : Type w) [Category.{w'} J] (F : C ⥤ D) :
    Subsingleton (PreservesColimitsOfShape J F) := by
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J✝ : Type w
inst✝¹ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
inst✝ : Category.{w', w} J
F : C ⥤ D
⊢ Subsingleton (PreservesColimitsOfShape J F)

Next tactic:
constructor
instance preservesColimitsOfShape_subsingleton (J : Type w) [Category.{w'} J] (F : C ⥤ D) :
    Subsingleton (PreservesColimitsOfShape J F) := by
  constructor; 

Proof state:
case allEq
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J✝ : Type w
inst✝¹ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
inst✝ : Category.{w', w} J
F : C ⥤ D
⊢ ∀ (a b : PreservesColimitsOfShape J F), a = b

Next tactic:
rintro ⟨a⟩ ⟨b⟩
instance preservesColimitsOfShape_subsingleton (J : Type w) [Category.{w'} J] (F : C ⥤ D) :
    Subsingleton (PreservesColimitsOfShape J F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; 

Proof state:
case allEq.mk.mk
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J✝ : Type w
inst✝¹ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
inst✝ : Category.{w', w} J
F : C ⥤ D
a b : {K : J ⥤ C} → PreservesColimit K F
⊢ PreservesColimitsOfShape.mk = PreservesColimitsOfShape.mk

Next tactic:
congr!
instance preserves_limits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesLimitsOfSize.{w', w} F) := by
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
⊢ Subsingleton (PreservesLimitsOfSize.{w', w, v₁, v₂, u₁, u₂} F)

Next tactic:
constructor
instance preserves_limits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesLimitsOfSize.{w', w} F) := by
  constructor; 

Proof state:
case allEq
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
⊢ ∀ (a b : PreservesLimitsOfSize.{w', w, v₁, v₂, u₁, u₂} F), a = b

Next tactic:
rintro ⟨a⟩ ⟨b⟩
instance preserves_limits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesLimitsOfSize.{w', w} F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; 

Proof state:
case allEq.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
a b : {J : Type w} → [inst : Category.{w', w} J] → PreservesLimitsOfShape J F
⊢ PreservesLimitsOfSize.mk = PreservesLimitsOfSize.mk

Next tactic:
congr
instance preserves_limits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesLimitsOfSize.{w', w} F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; congr; 

Proof state:
case allEq.mk.mk.e_preservesLimitsOfShape
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
a b : {J : Type w} → [inst : Category.{w', w} J] → PreservesLimitsOfShape J F
⊢ a = b

Next tactic:
funext
instance preserves_limits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesLimitsOfSize.{w', w} F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; congr; funext; 

Proof state:
case allEq.mk.mk.e_preservesLimitsOfShape.h.h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
a b : {J : Type w} → [inst : Category.{w', w} J] → PreservesLimitsOfShape J F
x✝¹ : Type w
x✝ : Category.{w', w} x✝¹
⊢ a = b

Next tactic:
congr!
instance preserves_colimits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesColimitsOfSize.{w', w} F) := by
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
⊢ Subsingleton (PreservesColimitsOfSize.{w', w, v₁, v₂, u₁, u₂} F)

Next tactic:
constructor
instance preserves_colimits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesColimitsOfSize.{w', w} F) := by
  constructor; 

Proof state:
case allEq
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
⊢ ∀ (a b : PreservesColimitsOfSize.{w', w, v₁, v₂, u₁, u₂} F), a = b

Next tactic:
rintro ⟨a⟩ ⟨b⟩
instance preserves_colimits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesColimitsOfSize.{w', w} F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; 

Proof state:
case allEq.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
a b : {J : Type w} → [inst : Category.{w', w} J] → PreservesColimitsOfShape J F
⊢ PreservesColimitsOfSize.mk = PreservesColimitsOfSize.mk

Next tactic:
congr
instance preserves_colimits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesColimitsOfSize.{w', w} F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; congr; 

Proof state:
case allEq.mk.mk.e_preservesColimitsOfShape
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
a b : {J : Type w} → [inst : Category.{w', w} J] → PreservesColimitsOfShape J F
⊢ a = b

Next tactic:
funext
instance preserves_colimits_subsingleton (F : C ⥤ D) :
    Subsingleton (PreservesColimitsOfSize.{w', w} F) := by
  constructor; rintro ⟨a⟩ ⟨b⟩; congr; funext; 

Proof state:
case allEq.mk.mk.e_preservesColimitsOfShape.h.h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type w
inst✝ : Category.{w', w} J
K : J ⥤ C
F : C ⥤ D
a b : {J : Type w} → [inst : Category.{w', w} J] → PreservesColimitsOfShape J F
x✝¹ : Type w
x✝ : Category.{w', w} x✝¹
⊢ a = b

Next tactic:
congr!
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K✝ : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
K : J ⥤ C
c : Cone K
h : IsLimit c
⊢ ∀ (s : Cone (K ⋙ 𝟭 C)) (j : J),
    (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) s ≫ ((𝟭 C).mapCone c).π.app j =
      s.π.app j

Next tactic:
cases K
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            cases K; 

Proof state:
case mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
toPrefunctor✝ : J ⥤q C
map_id✝ : ∀ (X : J), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
c : Cone (Functor.mk toPrefunctor✝)
h : IsLimit c
⊢ ∀ (s : Cone (Functor.mk toPrefunctor✝ ⋙ 𝟭 C)) (j : J),
    (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) s ≫ ((𝟭 C).mapCone c).π.app j =
      s.π.app j

Next tactic:
rcases c with ⟨_, _, _⟩
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            cases K; rcases c with ⟨_, _, _⟩; 

Proof state:
case mk.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
toPrefunctor✝ : J ⥤q C
map_id✝ : ∀ (X : J), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
pt✝ : C
app✝ : (X : J) → ((Functor.const J).obj pt✝).obj X ⟶ (Functor.mk toPrefunctor✝).obj X
naturality✝ :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y), ((Functor.const J).obj pt✝).map f ≫ app✝ Y = app✝ X ≫ (Functor.mk toPrefunctor✝).map f
h : IsLimit { pt := pt✝, π := NatTrans.mk app✝ }
⊢ ∀ (s : Cone (Functor.mk toPrefunctor✝ ⋙ 𝟭 C)) (j : J),
    (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) s ≫
        ((𝟭 C).mapCone { pt := pt✝, π := NatTrans.mk app✝ }).π.app j =
      s.π.app j

Next tactic:
intro s j
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            cases K; rcases c with ⟨_, _, _⟩; intro s j; 

Proof state:
case mk.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
toPrefunctor✝ : J ⥤q C
map_id✝ : ∀ (X : J), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
pt✝ : C
app✝ : (X : J) → ((Functor.const J).obj pt✝).obj X ⟶ (Functor.mk toPrefunctor✝).obj X
naturality✝ :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y), ((Functor.const J).obj pt✝).map f ≫ app✝ Y = app✝ X ≫ (Functor.mk toPrefunctor✝).map f
h : IsLimit { pt := pt✝, π := NatTrans.mk app✝ }
s : Cone (Functor.mk toPrefunctor✝ ⋙ 𝟭 C)
j : J
⊢ (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) s ≫
      ((𝟭 C).mapCone { pt := pt✝, π := NatTrans.mk app✝ }).π.app j =
    s.π.app j

Next tactic:
cases s
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            cases K; rcases c with ⟨_, _, _⟩; intro s j; cases s; 

Proof state:
case mk.mk.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
toPrefunctor✝ : J ⥤q C
map_id✝ : ∀ (X : J), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
pt✝¹ : C
app✝ : (X : J) → ((Functor.const J).obj pt✝¹).obj X ⟶ (Functor.mk toPrefunctor✝).obj X
naturality✝ :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y), ((Functor.const J).obj pt✝¹).map f ≫ app✝ Y = app✝ X ≫ (Functor.mk toPrefunctor✝).map f
h : IsLimit { pt := pt✝¹, π := NatTrans.mk app✝ }
j : J
pt✝ : C
π✝ : (Functor.const J).obj pt✝ ⟶ Functor.mk toPrefunctor✝ ⋙ 𝟭 C
⊢ (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) { pt := pt✝, π := π✝ } ≫
      ((𝟭 C).mapCone { pt := pt✝¹, π := NatTrans.mk app✝ }).π.app j =
    { pt := pt✝, π := π✝ }.π.app j

Next tactic:
exact h.fac _ j
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            cases K; rcases c with ⟨_, _, _⟩; intro s j; cases s; exact h.fac _ j, by
            

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K✝ : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
K : J ⥤ C
c : Cone K
h : IsLimit c
⊢ ∀ (s : Cone (K ⋙ 𝟭 C)) (m : s.pt ⟶ ((𝟭 C).mapCone c).pt),
    (∀ (j : J), m ≫ ((𝟭 C).mapCone c).π.app j = s.π.app j) →
      m = (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) s

Next tactic:
cases K
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            cases K; rcases c with ⟨_, _, _⟩; intro s j; cases s; exact h.fac _ j, by
            cases K; 

Proof state:
case mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
toPrefunctor✝ : J ⥤q C
map_id✝ : ∀ (X : J), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
c : Cone (Functor.mk toPrefunctor✝)
h : IsLimit c
⊢ ∀ (s : Cone (Functor.mk toPrefunctor✝ ⋙ 𝟭 C)) (m : s.pt ⟶ ((𝟭 C).mapCone c).pt),
    (∀ (j : J), m ≫ ((𝟭 C).mapCone c).π.app j = s.π.app j) →
      m = (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) s

Next tactic:
rcases c with ⟨_, _, _⟩
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            cases K; rcases c with ⟨_, _, _⟩; intro s j; cases s; exact h.fac _ j, by
            cases K; rcases c with ⟨_, _, _⟩; 

Proof state:
case mk.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
toPrefunctor✝ : J ⥤q C
map_id✝ : ∀ (X : J), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
pt✝ : C
app✝ : (X : J) → ((Functor.const J).obj pt✝).obj X ⟶ (Functor.mk toPrefunctor✝).obj X
naturality✝ :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y), ((Functor.const J).obj pt✝).map f ≫ app✝ Y = app✝ X ≫ (Functor.mk toPrefunctor✝).map f
h : IsLimit { pt := pt✝, π := NatTrans.mk app✝ }
⊢ ∀ (s : Cone (Functor.mk toPrefunctor✝ ⋙ 𝟭 C)) (m : s.pt ⟶ ((𝟭 C).mapCone { pt := pt✝, π := NatTrans.mk app✝ }).pt),
    (∀ (j : J), m ≫ ((𝟭 C).mapCone { pt := pt✝, π := NatTrans.mk app✝ }).π.app j = s.π.app j) →
      m = (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) s

Next tactic:
intro s m w
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            cases K; rcases c with ⟨_, _, _⟩; intro s j; cases s; exact h.fac _ j, by
            cases K; rcases c with ⟨_, _, _⟩; intro s m w; 

Proof state:
case mk.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
toPrefunctor✝ : J ⥤q C
map_id✝ : ∀ (X : J), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
pt✝ : C
app✝ : (X : J) → ((Functor.const J).obj pt✝).obj X ⟶ (Functor.mk toPrefunctor✝).obj X
naturality✝ :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y), ((Functor.const J).obj pt✝).map f ≫ app✝ Y = app✝ X ≫ (Functor.mk toPrefunctor✝).map f
h : IsLimit { pt := pt✝, π := NatTrans.mk app✝ }
s : Cone (Functor.mk toPrefunctor✝ ⋙ 𝟭 C)
m : s.pt ⟶ ((𝟭 C).mapCone { pt := pt✝, π := NatTrans.mk app✝ }).pt
w : ∀ (j : J), m ≫ ((𝟭 C).mapCone { pt := pt✝, π := NatTrans.mk app✝ }).π.app j = s.π.app j
⊢ m = (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) s

Next tactic:
rcases s with ⟨_, _, _⟩
instance idPreservesLimits : PreservesLimitsOfSize.{w', w} (𝟭 C) where
  preservesLimitsOfShape {J} 𝒥 :=
    {
      preservesLimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.lift ⟨s.pt, fun j => s.π.app j, fun j j' f => s.π.naturality f⟩, by
            cases K; rcases c with ⟨_, _, _⟩; intro s j; cases s; exact h.fac _ j, by
            cases K; rcases c with ⟨_, _, _⟩; intro s m w; rcases s with ⟨_, _, _⟩;
              

Proof state:
case mk.mk.mk.mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
toPrefunctor✝ : J ⥤q C
map_id✝ : ∀ (X : J), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
pt✝¹ : C
app✝¹ : (X : J) → ((Functor.const J).obj pt✝¹).obj X ⟶ (Functor.mk toPrefunctor✝).obj X
naturality✝¹ :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y), ((Functor.const J).obj pt✝¹).map f ≫ app✝¹ Y = app✝¹ X ≫ (Functor.mk toPrefunctor✝).map f
h : IsLimit { pt := pt✝¹, π := NatTrans.mk app✝¹ }
pt✝ : C
app✝ : (X : J) → ((Functor.const J).obj pt✝).obj X ⟶ (Functor.mk toPrefunctor✝ ⋙ 𝟭 C).obj X
naturality✝ :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y), ((Functor.const J).obj pt✝).map f ≫ app✝ Y = app✝ X ≫ (Functor.mk toPrefunctor✝ ⋙ 𝟭 C).map f
m : { pt := pt✝, π := NatTrans.mk app✝ }.pt ⟶ ((𝟭 C).mapCone { pt := pt✝¹, π := NatTrans.mk app✝¹ }).pt
w :
  ∀ (j : J),
    m ≫ ((𝟭 C).mapCone { pt := pt✝¹, π := NatTrans.mk app✝¹ }).π.app j = { pt := pt✝, π := NatTrans.mk app✝ }.π.app j
⊢ m = (fun s => IsLimit.lift h { pt := s.pt, π := NatTrans.mk fun j => s.π.app j }) { pt := pt✝, π := NatTrans.mk app✝ }

Next tactic:
exact h.uniq _ m w
instance idPreservesColimits : PreservesColimitsOfSize.{w', w} (𝟭 C) where
  preservesColimitsOfShape {J} 𝒥 :=
    {
      preservesColimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.desc ⟨s.pt, fun j => s.ι.app j, fun j j' f => s.ι.naturality f⟩, by
            

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K✝ : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
K : J ⥤ C
c : Cocone K
h : IsColimit c
⊢ ∀ (s : Cocone (K ⋙ 𝟭 C)) (j : J),
    ((𝟭 C).mapCocone c).ι.app j ≫ (fun s => IsColimit.desc h { pt := s.pt, ι := NatTrans.mk fun j => s.ι.app j }) s =
      s.ι.app j

Next tactic:
cases K
instance idPreservesColimits : PreservesColimitsOfSize.{w', w} (𝟭 C) where
  preservesColimitsOfShape {J} 𝒥 :=
    {
      preservesColimit := fun {K} =>
        ⟨fun {c} h =>
          ⟨fun s => h.desc ⟨s.pt, fun j => s.ι.app j, fun j j' f => s.ι.naturality f⟩, by
            cases K; 

Proof state:
case mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ C
J : Type w
𝒥 : Category.{w', w} J
toPrefunctor✝ : J ⥤q C
map_id✝ : ∀ (X : J), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
c : Cocone (Functor.mk toPrefunctor✝)
h : IsColimit c
⊢ ∀ (s : Cocone (Functor.mk toPrefunctor✝ ⋙ 𝟭 C)) (j : J),
    ((𝟭 C).mapCocone c).ι.app j ≫ (fun s => IsColimit.desc h { pt := s.pt, ι := NatTrans.mk fun j => s.ι.app j }) s =
      s.ι.app j

Next tactic:
rcases c with ⟨_, _, _⟩