l3lab/ntp-mathlib huggingface dataset:
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
    use StructuredArrow.homMk Z''.right (by simp [← G.map_comp])
  · intro Y Z f g
    let W :=
      @IsCofiltered.eq (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom) (StructuredArrow.homMk (F.map f.right) (StructuredArrow.w f))
        (StructuredArrow.homMk (F.map g.right) (StructuredArrow.w g))
    let h : W ⟶ _ := IsCofiltered.eqHom _ _
    

Proof state:
case this.cone_maps
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
f g : Y ⟶ Z
W : StructuredArrow X G :=
  IsCofiltered.eq (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
h : W ⟶ StructuredArrow.mk Y.hom :=
  IsCofiltered.eqHom (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
⊢ ∃ W h, h ≫ f = h ≫ g

Next tactic:
let h_cond : h ≫ _ = h ≫ _ := IsCofiltered.eq_condition _ _
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
    use StructuredArrow.homMk Z''.right (by simp [← G.map_comp])
  · intro Y Z f g
    let W :=
      @IsCofiltered.eq (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom) (StructuredArrow.homMk (F.map f.right) (StructuredArrow.w f))
        (StructuredArrow.homMk (F.map g.right) (StructuredArrow.w g))
    let h : W ⟶ _ := IsCofiltered.eqHom _ _
    let h_cond : h ≫ _ = h ≫ _ := IsCofiltered.eq_condition _ _
    

Proof state:
case this.cone_maps
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
f g : Y ⟶ Z
W : StructuredArrow X G :=
  IsCofiltered.eq (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
h : W ⟶ StructuredArrow.mk Y.hom :=
  IsCofiltered.eqHom (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
h_cond : h ≫ StructuredArrow.homMk (F.map f.right) = h ≫ StructuredArrow.homMk (F.map g.right) :=
  IsCofiltered.eq_condition (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
⊢ ∃ W h, h ≫ f = h ≫ g

Next tactic:
let W' :=
      @IsCofiltered.eq (StructuredArrow W.right F) _ _ (StructuredArrow.mk h.right)
        (StructuredArrow.mk (h.right ≫ F.map f.right)) (StructuredArrow.homMk f.right rfl)
        (StructuredArrow.homMk g.right (congr_arg CommaMorphism.right h_cond).symm)
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
    use StructuredArrow.homMk Z''.right (by simp [← G.map_comp])
  · intro Y Z f g
    let W :=
      @IsCofiltered.eq (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom) (StructuredArrow.homMk (F.map f.right) (StructuredArrow.w f))
        (StructuredArrow.homMk (F.map g.right) (StructuredArrow.w g))
    let h : W ⟶ _ := IsCofiltered.eqHom _ _
    let h_cond : h ≫ _ = h ≫ _ := IsCofiltered.eq_condition _ _
    let W' :=
      @IsCofiltered.eq (StructuredArrow W.right F) _ _ (StructuredArrow.mk h.right)
        (StructuredArrow.mk (h.right ≫ F.map f.right)) (StructuredArrow.homMk f.right rfl)
        (StructuredArrow.homMk g.right (congr_arg CommaMorphism.right h_cond).symm)
    

Proof state:
case this.cone_maps
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
f g : Y ⟶ Z
W : StructuredArrow X G :=
  IsCofiltered.eq (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
h : W ⟶ StructuredArrow.mk Y.hom :=
  IsCofiltered.eqHom (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
h_cond : h ≫ StructuredArrow.homMk (F.map f.right) = h ≫ StructuredArrow.homMk (F.map g.right) :=
  IsCofiltered.eq_condition (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
W' : StructuredArrow W.right F := IsCofiltered.eq (StructuredArrow.homMk f.right) (StructuredArrow.homMk g.right)
⊢ ∃ W h, h ≫ f = h ≫ g

Next tactic:
let h' : W' ⟶ _ := IsCofiltered.eqHom _ _
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
    use StructuredArrow.homMk Z''.right (by simp [← G.map_comp])
  · intro Y Z f g
    let W :=
      @IsCofiltered.eq (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom) (StructuredArrow.homMk (F.map f.right) (StructuredArrow.w f))
        (StructuredArrow.homMk (F.map g.right) (StructuredArrow.w g))
    let h : W ⟶ _ := IsCofiltered.eqHom _ _
    let h_cond : h ≫ _ = h ≫ _ := IsCofiltered.eq_condition _ _
    let W' :=
      @IsCofiltered.eq (StructuredArrow W.right F) _ _ (StructuredArrow.mk h.right)
        (StructuredArrow.mk (h.right ≫ F.map f.right)) (StructuredArrow.homMk f.right rfl)
        (StructuredArrow.homMk g.right (congr_arg CommaMorphism.right h_cond).symm)
    let h' : W' ⟶ _ := IsCofiltered.eqHom _ _
    

Proof state:
case this.cone_maps
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
f g : Y ⟶ Z
W : StructuredArrow X G :=
  IsCofiltered.eq (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
h : W ⟶ StructuredArrow.mk Y.hom :=
  IsCofiltered.eqHom (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
h_cond : h ≫ StructuredArrow.homMk (F.map f.right) = h ≫ StructuredArrow.homMk (F.map g.right) :=
  IsCofiltered.eq_condition (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
W' : StructuredArrow W.right F := IsCofiltered.eq (StructuredArrow.homMk f.right) (StructuredArrow.homMk g.right)
h' : W' ⟶ StructuredArrow.mk h.right :=
  IsCofiltered.eqHom (StructuredArrow.homMk f.right) (StructuredArrow.homMk g.right)
⊢ ∃ W h, h ≫ f = h ≫ g

Next tactic:
let h'_cond : h' ≫ _ = h' ≫ _ := IsCofiltered.eq_condition _ _
theorem flat_of_preservesFiniteLimits [HasFiniteLimits C] (F : C ⥤ D) [PreservesFiniteLimits F] :
    RepresentablyFlat F :=
  ⟨fun X =>
    haveI : HasFiniteLimits (StructuredArrow X F) := by
      

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₁, u₂} D
inst✝¹ : HasFiniteLimits C
F : C ⥤ D
inst✝ : PreservesFiniteLimits F
X : D
⊢ HasFiniteLimits (StructuredArrow X F)

Next tactic:
apply hasFiniteLimits_of_hasFiniteLimits_of_size.{v₁} (StructuredArrow X F)
theorem flat_of_preservesFiniteLimits [HasFiniteLimits C] (F : C ⥤ D) [PreservesFiniteLimits F] :
    RepresentablyFlat F :=
  ⟨fun X =>
    haveI : HasFiniteLimits (StructuredArrow X F) := by
      apply hasFiniteLimits_of_hasFiniteLimits_of_size.{v₁} (StructuredArrow X F)
      

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₁, u₂} D
inst✝¹ : HasFiniteLimits C
F : C ⥤ D
inst✝ : PreservesFiniteLimits F
X : D
⊢ ∀ (J : Type v₁) {𝒥 : SmallCategory J}, FinCategory J → HasLimitsOfShape J (StructuredArrow X F)

Next tactic:
intro J sJ fJ
theorem flat_of_preservesFiniteLimits [HasFiniteLimits C] (F : C ⥤ D) [PreservesFiniteLimits F] :
    RepresentablyFlat F :=
  ⟨fun X =>
    haveI : HasFiniteLimits (StructuredArrow X F) := by
      apply hasFiniteLimits_of_hasFiniteLimits_of_size.{v₁} (StructuredArrow X F)
      intro J sJ fJ
      

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₁, u₂} D
inst✝¹ : HasFiniteLimits C
F : C ⥤ D
inst✝ : PreservesFiniteLimits F
X : D
J : Type v₁
sJ : SmallCategory J
fJ : FinCategory J
⊢ HasLimitsOfShape J (StructuredArrow X F)

Next tactic:
constructor
theorem flat_of_preservesFiniteLimits [HasFiniteLimits C] (F : C ⥤ D) [PreservesFiniteLimits F] :
    RepresentablyFlat F :=
  ⟨fun X =>
    haveI : HasFiniteLimits (StructuredArrow X F) := by
      apply hasFiniteLimits_of_hasFiniteLimits_of_size.{v₁} (StructuredArrow X F)
      intro J sJ fJ
      constructor
      -- porting note: instance was inferred automatically in Lean 3
      

Proof state:
case has_limit
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₁, u₂} D
inst✝¹ : HasFiniteLimits C
F : C ⥤ D
inst✝ : PreservesFiniteLimits F
X : D
J : Type v₁
sJ : SmallCategory J
fJ : FinCategory J
⊢ autoParam (∀ (F_1 : J ⥤ StructuredArrow X F), HasLimit F_1) _auto✝

Next tactic:
infer_instance
theorem fac (x : J) : lift F hc s ≫ (F.mapCone c).π.app x = s.π.app x := by
  

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₁, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K : J ⥤ C
F : C ⥤ D
inst✝ : RepresentablyFlat F
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
x : J
⊢ lift F hc s ≫ (F.mapCone c).π.app x = s.π.app x

Next tactic:
simp [lift, ← Functor.map_comp]
theorem uniq {K : J ⥤ C} {c : Cone K} (hc : IsLimit c) (s : Cone (K ⋙ F))
    (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ j : J, f₁ ≫ (F.mapCone c).π.app j = s.π.app j)
    (h₂ : ∀ j : J, f₂ ≫ (F.mapCone c).π.app j = s.π.app j) : f₁ = f₂ := by
  -- We can make two cones over the diagram of `s` via `f₁` and `f₂`.
  

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₁, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K✝ : J ⥤ C
F : C ⥤ D
inst✝ : RepresentablyFlat F
c✝ : Cone K✝
hc✝ : IsLimit c✝
s✝ : Cone (K✝ ⋙ F)
K : J ⥤ C
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
f₁ f₂ : s.pt ⟶ F.obj c.pt
h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j
h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j
⊢ f₁ = f₂

Next tactic:
let α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₁]) }
theorem uniq {K : J ⥤ C} {c : Cone K} (hc : IsLimit c) (s : Cone (K ⋙ F))
    (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ j : J, f₁ ≫ (F.mapCone c).π.app j = s.π.app j)
    (h₂ : ∀ j : J, f₂ ≫ (F.mapCone c).π.app j = s.π.app j) : f₁ = f₂ := by
  -- We can make two cones over the diagram of `s` via `f₁` and `f₂`.
  let α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by 

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₁, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K✝ : J ⥤ C
F : C ⥤ D
inst✝ : RepresentablyFlat F
c✝ : Cone K✝
hc✝ : IsLimit c✝
s✝ : Cone (K✝ ⋙ F)
K : J ⥤ C
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
f₁ f₂ : s.pt ⟶ F.obj c.pt
h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j
h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j
X : J
⊢ (Cone.toStructuredArrow (F.mapCone c) ⋙ map f₁).obj X = (Cone.toStructuredArrow s).obj X

Next tactic:
simp [← h₁]
theorem uniq {K : J ⥤ C} {c : Cone K} (hc : IsLimit c) (s : Cone (K ⋙ F))
    (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ j : J, f₁ ≫ (F.mapCone c).π.app j = s.π.app j)
    (h₂ : ∀ j : J, f₂ ≫ (F.mapCone c).π.app j = s.π.app j) : f₁ = f₂ := by
  -- We can make two cones over the diagram of `s` via `f₁` and `f₂`.
  let α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₁]) }
  

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₁, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K✝ : J ⥤ C
F : C ⥤ D
inst✝ : RepresentablyFlat F
c✝ : Cone K✝
hc✝ : IsLimit c✝
s✝ : Cone (K✝ ⋙ F)
K : J ⥤ C
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
f₁ f₂ : s.pt ⟶ F.obj c.pt
h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j
h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j
α₁ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₁ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₁ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
⊢ f₁ = f₂

Next tactic:
let α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₂]) }
theorem uniq {K : J ⥤ C} {c : Cone K} (hc : IsLimit c) (s : Cone (K ⋙ F))
    (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ j : J, f₁ ≫ (F.mapCone c).π.app j = s.π.app j)
    (h₂ : ∀ j : J, f₂ ≫ (F.mapCone c).π.app j = s.π.app j) : f₁ = f₂ := by
  -- We can make two cones over the diagram of `s` via `f₁` and `f₂`.
  let α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₁]) }
  let α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by 

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₁, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K✝ : J ⥤ C
F : C ⥤ D
inst✝ : RepresentablyFlat F
c✝ : Cone K✝
hc✝ : IsLimit c✝
s✝ : Cone (K✝ ⋙ F)
K : J ⥤ C
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
f₁ f₂ : s.pt ⟶ F.obj c.pt
h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j
h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j
α₁ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₁ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₁ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
X : J
⊢ (Cone.toStructuredArrow (F.mapCone c) ⋙ map f₂).obj X = (Cone.toStructuredArrow s).obj X

Next tactic:
simp [← h₂]
theorem uniq {K : J ⥤ C} {c : Cone K} (hc : IsLimit c) (s : Cone (K ⋙ F))
    (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ j : J, f₁ ≫ (F.mapCone c).π.app j = s.π.app j)
    (h₂ : ∀ j : J, f₂ ≫ (F.mapCone c).π.app j = s.π.app j) : f₁ = f₂ := by
  -- We can make two cones over the diagram of `s` via `f₁` and `f₂`.
  let α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₁]) }
  let α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₂]) }
  

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₁, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K✝ : J ⥤ C
F : C ⥤ D
inst✝ : RepresentablyFlat F
c✝ : Cone K✝
hc✝ : IsLimit c✝
s✝ : Cone (K✝ ⋙ F)
K : J ⥤ C
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
f₁ f₂ : s.pt ⟶ F.obj c.pt
h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j
h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j
α₁ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₁ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₁ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
α₂ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₂ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₂ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
⊢ f₁ = f₂

Next tactic:
let c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
    (Cones.postcompose (whiskerRight α₁ (pre s.pt K F) : _)).obj (c.toStructuredArrowCone F f₁)
theorem uniq {K : J ⥤ C} {c : Cone K} (hc : IsLimit c) (s : Cone (K ⋙ F))
    (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ j : J, f₁ ≫ (F.mapCone c).π.app j = s.π.app j)
    (h₂ : ∀ j : J, f₂ ≫ (F.mapCone c).π.app j = s.π.app j) : f₁ = f₂ := by
  -- We can make two cones over the diagram of `s` via `f₁` and `f₂`.
  let α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₁]) }
  let α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₂]) }
  let c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
    (Cones.postcompose (whiskerRight α₁ (pre s.pt K F) : _)).obj (c.toStructuredArrowCone F f₁)
  

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₁, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K✝ : J ⥤ C
F : C ⥤ D
inst✝ : RepresentablyFlat F
c✝ : Cone K✝
hc✝ : IsLimit c✝
s✝ : Cone (K✝ ⋙ F)
K : J ⥤ C
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
f₁ f₂ : s.pt ⟶ F.obj c.pt
h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j
h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j
α₁ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₁ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₁ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
α₂ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₂ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₂ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
c₁ : Cone (Cone.toStructuredArrow s ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₁ (pre s.pt K F))).obj (Cone.toStructuredArrowCone c F f₁)
⊢ f₁ = f₂

Next tactic:
let c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
    (Cones.postcompose (whiskerRight α₂ (pre s.pt K F) : _)).obj (c.toStructuredArrowCone F f₂)
theorem uniq {K : J ⥤ C} {c : Cone K} (hc : IsLimit c) (s : Cone (K ⋙ F))
    (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ j : J, f₁ ≫ (F.mapCone c).π.app j = s.π.app j)
    (h₂ : ∀ j : J, f₂ ≫ (F.mapCone c).π.app j = s.π.app j) : f₁ = f₂ := by
  -- We can make two cones over the diagram of `s` via `f₁` and `f₂`.
  let α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₁]) }
  let α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₂]) }
  let c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
    (Cones.postcompose (whiskerRight α₁ (pre s.pt K F) : _)).obj (c.toStructuredArrowCone F f₁)
  let c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
    (Cones.postcompose (whiskerRight α₂ (pre s.pt K F) : _)).obj (c.toStructuredArrowCone F f₂)
  -- The two cones can then be combined and we may obtain a cone over the two cones since
  -- `StructuredArrow s.pt F` is cofiltered.
  

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₁, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K✝ : J ⥤ C
F : C ⥤ D
inst✝ : RepresentablyFlat F
c✝ : Cone K✝
hc✝ : IsLimit c✝
s✝ : Cone (K✝ ⋙ F)
K : J ⥤ C
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
f₁ f₂ : s.pt ⟶ F.obj c.pt
h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j
h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j
α₁ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₁ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₁ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
α₂ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₂ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₂ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
c₁ : Cone (Cone.toStructuredArrow s ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₁ (pre s.pt K F))).obj (Cone.toStructuredArrowCone c F f₁)
c₂ : Cone (Cone.toStructuredArrow s ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₂ (pre s.pt K F))).obj (Cone.toStructuredArrowCone c F f₂)
⊢ f₁ = f₂

Next tactic:
let c₀ := IsCofiltered.cone (biconeMk _ c₁ c₂)