l3lab/ntp-mathlib huggingface dataset:
theorem yonedaFamily_fromCocone_compatible (S : Sieve X) (s : Cocone (diagram S.arrows)) :
    FamilyOfElements.Compatible <| yonedaFamilyOfElements_fromCocone S.arrows s := by
  intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ hgf
  have Hs := s.ι.naturality
  simp only [Functor.id_obj, yoneda_obj_obj, Opposite.unop_op, yoneda_obj_map, Quiver.Hom.unop_op]
  dsimp [yonedaFamilyOfElements_fromCocone]
  have hgf₁ : S.arrows (g₁ ≫ f₁) := by exact Sieve.downward_closed S hf₁ g₁
  have hgf₂ : S.arrows (g₂ ≫ f₂) := by exact Sieve.downward_closed S hf₂ g₂
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
Hs :
  ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
    (diagram S.arrows).map f ≫ s.ι.app Y =
      s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }

Next tactic:
let F : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk (g₂ ≫ f₂) : Over X) := (Over.homMk (𝟙 Z) )
theorem yonedaFamily_fromCocone_compatible (S : Sieve X) (s : Cocone (diagram S.arrows)) :
    FamilyOfElements.Compatible <| yonedaFamilyOfElements_fromCocone S.arrows s := by
  intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ hgf
  have Hs := s.ι.naturality
  simp only [Functor.id_obj, yoneda_obj_obj, Opposite.unop_op, yoneda_obj_map, Quiver.Hom.unop_op]
  dsimp [yonedaFamilyOfElements_fromCocone]
  have hgf₁ : S.arrows (g₁ ≫ f₁) := by exact Sieve.downward_closed S hf₁ g₁
  have hgf₂ : S.arrows (g₂ ≫ f₂) := by exact Sieve.downward_closed S hf₂ g₂
  let F : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk (g₂ ≫ f₂) : Over X) := (Over.homMk (𝟙 Z) )
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
Hs :
  ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
    (diagram S.arrows).map f ≫ s.ι.app Y =
      s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }

Next tactic:
let F₁ : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk f₁ : Over X) := (Over.homMk g₁)
theorem yonedaFamily_fromCocone_compatible (S : Sieve X) (s : Cocone (diagram S.arrows)) :
    FamilyOfElements.Compatible <| yonedaFamilyOfElements_fromCocone S.arrows s := by
  intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ hgf
  have Hs := s.ι.naturality
  simp only [Functor.id_obj, yoneda_obj_obj, Opposite.unop_op, yoneda_obj_map, Quiver.Hom.unop_op]
  dsimp [yonedaFamilyOfElements_fromCocone]
  have hgf₁ : S.arrows (g₁ ≫ f₁) := by exact Sieve.downward_closed S hf₁ g₁
  have hgf₂ : S.arrows (g₂ ≫ f₂) := by exact Sieve.downward_closed S hf₂ g₂
  let F : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk (g₂ ≫ f₂) : Over X) := (Over.homMk (𝟙 Z) )
  let F₁ : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk f₁ : Over X) := (Over.homMk g₁)
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
Hs :
  ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
    (diagram S.arrows).map f ≫ s.ι.app Y =
      s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }

Next tactic:
let F₂ : (Over.mk (g₂ ≫ f₂) : Over X) ⟶ (Over.mk f₂ : Over X) := (Over.homMk g₂)
theorem yonedaFamily_fromCocone_compatible (S : Sieve X) (s : Cocone (diagram S.arrows)) :
    FamilyOfElements.Compatible <| yonedaFamilyOfElements_fromCocone S.arrows s := by
  intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ hgf
  have Hs := s.ι.naturality
  simp only [Functor.id_obj, yoneda_obj_obj, Opposite.unop_op, yoneda_obj_map, Quiver.Hom.unop_op]
  dsimp [yonedaFamilyOfElements_fromCocone]
  have hgf₁ : S.arrows (g₁ ≫ f₁) := by exact Sieve.downward_closed S hf₁ g₁
  have hgf₂ : S.arrows (g₂ ≫ f₂) := by exact Sieve.downward_closed S hf₂ g₂
  let F : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk (g₂ ≫ f₂) : Over X) := (Over.homMk (𝟙 Z) )
  let F₁ : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk f₁ : Over X) := (Over.homMk g₁)
  let F₂ : (Over.mk (g₂ ≫ f₂) : Over X) ⟶ (Over.mk f₂ : Over X) := (Over.homMk g₂)
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
Hs :
  ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
    (diagram S.arrows).map f ≫ s.ι.app Y =
      s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }

Next tactic:
have hF := @Hs ⟨Over.mk (g₁ ≫ f₁), hgf₁⟩ ⟨Over.mk (g₂ ≫ f₂), hgf₂⟩ F
theorem yonedaFamily_fromCocone_compatible (S : Sieve X) (s : Cocone (diagram S.arrows)) :
    FamilyOfElements.Compatible <| yonedaFamilyOfElements_fromCocone S.arrows s := by
  intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ hgf
  have Hs := s.ι.naturality
  simp only [Functor.id_obj, yoneda_obj_obj, Opposite.unop_op, yoneda_obj_map, Quiver.Hom.unop_op]
  dsimp [yonedaFamilyOfElements_fromCocone]
  have hgf₁ : S.arrows (g₁ ≫ f₁) := by exact Sieve.downward_closed S hf₁ g₁
  have hgf₂ : S.arrows (g₂ ≫ f₂) := by exact Sieve.downward_closed S hf₂ g₂
  let F : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk (g₂ ≫ f₂) : Over X) := (Over.homMk (𝟙 Z) )
  let F₁ : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk f₁ : Over X) := (Over.homMk g₁)
  let F₂ : (Over.mk (g₂ ≫ f₂) : Over X) ⟶ (Over.mk f₂ : Over X) := (Over.homMk g₂)
  have hF := @Hs ⟨Over.mk (g₁ ≫ f₁), hgf₁⟩ ⟨Over.mk (g₂ ≫ f₂), hgf₂⟩ F
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
Hs :
  ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
    (diagram S.arrows).map f ≫ s.ι.app Y =
      s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
  (diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
    s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
      ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }

Next tactic:
have hF₁ := @Hs ⟨Over.mk (g₁ ≫ f₁), hgf₁⟩ ⟨Over.mk f₁, hf₁⟩ F₁
theorem yonedaFamily_fromCocone_compatible (S : Sieve X) (s : Cocone (diagram S.arrows)) :
    FamilyOfElements.Compatible <| yonedaFamilyOfElements_fromCocone S.arrows s := by
  intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ hgf
  have Hs := s.ι.naturality
  simp only [Functor.id_obj, yoneda_obj_obj, Opposite.unop_op, yoneda_obj_map, Quiver.Hom.unop_op]
  dsimp [yonedaFamilyOfElements_fromCocone]
  have hgf₁ : S.arrows (g₁ ≫ f₁) := by exact Sieve.downward_closed S hf₁ g₁
  have hgf₂ : S.arrows (g₂ ≫ f₂) := by exact Sieve.downward_closed S hf₂ g₂
  let F : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk (g₂ ≫ f₂) : Over X) := (Over.homMk (𝟙 Z) )
  let F₁ : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk f₁ : Over X) := (Over.homMk g₁)
  let F₂ : (Over.mk (g₂ ≫ f₂) : Over X) ⟶ (Over.mk f₂ : Over X) := (Over.homMk g₂)
  have hF := @Hs ⟨Over.mk (g₁ ≫ f₁), hgf₁⟩ ⟨Over.mk (g₂ ≫ f₂), hgf₂⟩ F
  have hF₁ := @Hs ⟨Over.mk (g₁ ≫ f₁), hgf₁⟩ ⟨Over.mk f₁, hf₁⟩ F₁
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
Hs :
  ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
    (diagram S.arrows).map f ≫ s.ι.app Y =
      s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
  (diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
    s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
      ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
  (diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
    s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
      ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }

Next tactic:
have hF₂ := @Hs ⟨Over.mk (g₂ ≫ f₂), hgf₂⟩ ⟨Over.mk f₂, hf₂⟩ F₂
theorem yonedaFamily_fromCocone_compatible (S : Sieve X) (s : Cocone (diagram S.arrows)) :
    FamilyOfElements.Compatible <| yonedaFamilyOfElements_fromCocone S.arrows s := by
  intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ hgf
  have Hs := s.ι.naturality
  simp only [Functor.id_obj, yoneda_obj_obj, Opposite.unop_op, yoneda_obj_map, Quiver.Hom.unop_op]
  dsimp [yonedaFamilyOfElements_fromCocone]
  have hgf₁ : S.arrows (g₁ ≫ f₁) := by exact Sieve.downward_closed S hf₁ g₁
  have hgf₂ : S.arrows (g₂ ≫ f₂) := by exact Sieve.downward_closed S hf₂ g₂
  let F : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk (g₂ ≫ f₂) : Over X) := (Over.homMk (𝟙 Z) )
  let F₁ : (Over.mk (g₁ ≫ f₁) : Over X) ⟶ (Over.mk f₁ : Over X) := (Over.homMk g₁)
  let F₂ : (Over.mk (g₂ ≫ f₂) : Over X) ⟶ (Over.mk f₂ : Over X) := (Over.homMk g₂)
  have hF := @Hs ⟨Over.mk (g₁ ≫ f₁), hgf₁⟩ ⟨Over.mk (g₂ ≫ f₂), hgf₂⟩ F
  have hF₁ := @Hs ⟨Over.mk (g₁ ≫ f₁), hgf₁⟩ ⟨Over.mk f₁, hf₁⟩ F₁
  have hF₂ := @Hs ⟨Over.mk (g₂ ≫ f₂), hgf₂⟩ ⟨Over.mk f₂, hf₂⟩ F₂
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
s : Cocone (diagram S.arrows)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : S.arrows f₁
hf₂ : S.arrows f₂
hgf : g₁ ≫ f₁ = g₂ ≫ f₂
Hs :
  ∀ ⦃X_1 Y : FullSubcategory fun f => S.arrows f.hom⦄ (f : X_1 ⟶ Y),
    (diagram S.arrows).map f ≫ s.ι.app Y =
      s.ι.app X_1 ≫ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f
hgf₁ : S.arrows (g₁ ≫ f₁)
hgf₂ : S.arrows (g₂ ≫ f₂)
F : Over.mk (g₁ ≫ f₁) ⟶ Over.mk (g₂ ≫ f₂) := Over.homMk (𝟙 Z)
F₁ : Over.mk (g₁ ≫ f₁) ⟶ Over.mk f₁ := Over.homMk g₁
F₂ : Over.mk (g₂ ≫ f₂) ⟶ Over.mk f₂ := Over.homMk g₂
hF :
  (diagram S.arrows).map F ≫ s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } =
    s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
      ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F
hF₁ :
  (diagram S.arrows).map F₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } =
    s.ι.app { obj := Over.mk (g₁ ≫ f₁), property := hgf₁ } ≫
      ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₁
hF₂ :
  (diagram S.arrows).map F₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ } =
    s.ι.app { obj := Over.mk (g₂ ≫ f₂), property := hgf₂ } ≫
      ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F₂
⊢ g₁ ≫ s.ι.app { obj := Over.mk f₁, property := hf₁ } = g₂ ≫ s.ι.app { obj := Over.mk f₂, property := hf₂ }

Next tactic:
aesop_cat
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) ↔ Nonempty (IsColimit (cocone S.arrows))

Next tactic:
constructor
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · 

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) → Nonempty (IsColimit (cocone S.arrows))

Next tactic:
intro H
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · intro H
    

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ Nonempty (IsColimit (cocone S.arrows))

Next tactic:
refine Nonempty.intro ?_
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · intro H
    refine Nonempty.intro ?_
    

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ IsColimit (cocone S.arrows)

Next tactic:
exact
    { desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
        (yonedaFamily_fromCocone_compatible S s) |>.choose
      fac := by
        intro s f
        replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
        have ht := H.choose_spec.1 f.obj.hom f.property
        aesop_cat
      uniq := by
        intro s Fs HFs
        replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
        apply H.choose_spec.2 Fs
        exact fun _ f hf => HFs ⟨Over.mk f, hf⟩ }
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · intro H
    refine Nonempty.intro ?_
    exact
    { desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
        (yonedaFamily_fromCocone_compatible S s) |>.choose
      fac := by
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ ∀ (s : Cocone (diagram S.arrows)) (j : FullSubcategory fun f => S.arrows f.hom),
    (cocone S.arrows).ι.app j ≫
        (fun s =>
            Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))
          s =
      s.ι.app j

Next tactic:
intro s f
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · intro H
    refine Nonempty.intro ?_
    exact
    { desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
        (yonedaFamily_fromCocone_compatible S s) |>.choose
      fac := by
        intro s f
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
⊢ (cocone S.arrows).ι.app f ≫
      (fun s =>
          Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))
        s =
    s.ι.app f

Next tactic:
replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · intro H
    refine Nonempty.intro ?_
    exact
    { desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
        (yonedaFamily_fromCocone_compatible S s) |>.choose
      fac := by
        intro s f
        replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t
⊢ (cocone S.arrows).ι.app f ≫
      (fun s =>
          Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))
        s =
    s.ι.app f

Next tactic:
have ht := H.choose_spec.1 f.obj.hom f.property
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · intro H
    refine Nonempty.intro ?_
    exact
    { desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
        (yonedaFamily_fromCocone_compatible S s) |>.choose
      fac := by
        intro s f
        replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
        have ht := H.choose_spec.1 f.obj.hom f.property
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
f : FullSubcategory fun f => S.arrows f.hom
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t
ht :
  (yoneda.obj s.pt).map f.obj.hom.op (Exists.choose H) =
    yonedaFamilyOfElements_fromCocone S.arrows s f.obj.hom (_ : S.arrows f.obj.hom)
⊢ (cocone S.arrows).ι.app f ≫
      (fun s =>
          Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))
        s =
    s.ι.app f

Next tactic:
aesop_cat
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · intro H
    refine Nonempty.intro ?_
    exact
    { desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
        (yonedaFamily_fromCocone_compatible S s) |>.choose
      fac := by
        intro s f
        replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
        have ht := H.choose_spec.1 f.obj.hom f.property
        aesop_cat
      uniq := by
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
⊢ ∀ (s : Cocone (diagram S.arrows)) (m : (cocone S.arrows).pt ⟶ s.pt),
    (∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ m = s.ι.app j) →
      m =
        (fun s =>
            Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))
          s

Next tactic:
intro s Fs HFs
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · intro H
    refine Nonempty.intro ?_
    exact
    { desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
        (yonedaFamily_fromCocone_compatible S s) |>.choose
      fac := by
        intro s f
        replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
        have ht := H.choose_spec.1 f.obj.hom f.property
        aesop_cat
      uniq := by
        intro s Fs HFs
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
H : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
⊢ Fs =
    (fun s =>
        Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))
      s

Next tactic:
replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
/--
The base of a sieve `S` is a colimit of `S` iff all Yoneda-presheaves satisfy
the sheaf condition for `S`.
-/
theorem forallYonedaIsSheaf_iff_colimit (S : Sieve X) :
    (∀ W : C, Presieve.IsSheafFor (yoneda.obj W) (S : Presieve X)) ↔
      Nonempty (IsColimit S.arrows.cocone) := by
  constructor
  · intro H
    refine Nonempty.intro ?_
    exact
    { desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
        (yonedaFamily_fromCocone_compatible S s) |>.choose
      fac := by
        intro s f
        replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
        have ht := H.choose_spec.1 f.obj.hom f.property
        aesop_cat
      uniq := by
        intro s Fs HFs
        replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
          (yonedaFamily_fromCocone_compatible S s)
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : C
S : Sieve X
H✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows
s : Cocone (diagram S.arrows)
Fs : (cocone S.arrows).pt ⟶ s.pt
HFs : ∀ (j : FullSubcategory fun f => S.arrows f.hom), (cocone S.arrows).ι.app j ≫ Fs = s.ι.app j
H : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t
⊢ Fs =
    (fun s =>
        Exists.choose (_ : ∃! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))
      s

Next tactic:
apply H.choose_spec.2 Fs