awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
a : J
𝒲 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
refine_2 :
  CommaMorphism.left
      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (d.gluedCocone (P := P).ι.app a ≫
          Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
            (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
              (fun (i : 𝒱.I₀) ↦
                CategoryStruct.comp (Y :=
                  (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                          (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                      (P := P) (Q := ⊤) (W := ⊤).left)
                  (CommaMorphism.left
                    (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                      ((d.isColimit (P := P) i).desc
                        (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
                  (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
              ⋯ ⋯)
            trivial trivial)) =
    (s.ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left
⊢ d.gluedCocone (P := P).ι.app a ≫
      Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
          (fun (i : 𝒱.I₀) ↦
            CategoryStruct.comp (Y :=
              (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                      (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                  (P := P) (Q := ⊤) (W := ⊤).left)
              (CommaMorphism.left
                (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                  ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
              (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
          ⋯ ⋯)
        trivial trivial =
    s.ι.app a

Tactic:
ext
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
a : J
𝒲 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
⊢ CommaMorphism.left
      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (d.gluedCocone (P := P).ι.app a ≫
          Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
            (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
              (fun (i : 𝒱.I₀) ↦
                CategoryStruct.comp (Y :=
                  (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                          (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                      (P := P) (Q := ⊤) (W := ⊤).left)
                  (CommaMorphism.left
                    (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                      ((d.isColimit (P := P) i).desc
                        (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
                  (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
              ⋯ ⋯)
            trivial trivial)) =
    (s.ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left

Tactic:
(𝒲 a).hom_ext _ _ fun i ↦ ?_

Premises used: AlgebraicGeometry.Scheme.Cover.hom_ext
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
a : J
𝒲 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
i : (𝒲 a).I₀
⊢ CategoryStruct.comp (Y :=
      Over.left ((Functor.obj (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s.pt).toComma (P := P) (Q := ⊤) (W := ⊤)))
      (Over.Hom.left
          (Comma.Hom.hom (P := P) (Q := ⊤) (W := ⊤)
            ((d.cocone (P := P) i).ι.app a ≫
              (d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))) :
        Over.left
            ((Functor.obj (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) (D.obj a)).toComma (P := P) (Q := ⊤) (W := ⊤)) ⟶
          Over.left
            ((Functor.obj (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s.pt).toComma (P := P) (Q := ⊤) (W := ⊤)))
      (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) :
        pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) ⟶ s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).left) =
    pullback.fst (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) ≫
      (s.ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left

Tactic:
simp
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
m : d.gluedCocone (P := P).pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j
refine_3 :
  m.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
    CommaMorphism.left
      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
          (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
            (fun (i : 𝒱.I₀) ↦
              CategoryStruct.comp (Y :=
                (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                        (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                    (P := P) (Q := ⊤) (W := ⊤).left)
                (CommaMorphism.left
                  (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                    ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
                (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
            ⋯ ⋯)
          trivial trivial))
⊢ m =
    Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
      (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
        (fun (i : 𝒱.I₀) ↦
          CategoryStruct.comp (Y :=
            (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                    (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                (P := P) (Q := ⊤) (W := ⊤).left)
            (CommaMorphism.left
              (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
            (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
        ⋯ ⋯)
      trivial trivial

Tactic:
ext
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
m : d.gluedCocone (P := P).pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j
⊢ m.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
    CommaMorphism.left
      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
          (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
            (fun (i : 𝒱.I₀) ↦
              CategoryStruct.comp (Y :=
                (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                        (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                    (P := P) (Q := ⊤) (W := ⊤).left)
                (CommaMorphism.left
                  (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                    ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
                (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
            ⋯ ⋯)
          trivial trivial))

Tactic:
𝒱.hom_ext _ _ fun i ↦ ?_

Premises used: AlgebraicGeometry.Scheme.Cover.hom_ext
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
m : d.gluedCocone (P := P).pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j
refine_3 :
  ∀ (i : 𝒱.I₀),
    𝒱.f i ≫ m.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
      𝒱.f i ≫
        CommaMorphism.left
          (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
            (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
              (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
                (fun (i : 𝒱.I₀) ↦
                  CategoryStruct.comp (Y :=
                    (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                            (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                        (P := P) (Q := ⊤) (W := ⊤).left)
                    (CommaMorphism.left
                      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                        ((d.isColimit (P := P) i).desc
                          (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
                    (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
                ⋯ ⋯)
              trivial trivial))
⊢ m.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
    CommaMorphism.left
      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
          (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
            (fun (i : 𝒱.I₀) ↦
              CategoryStruct.comp (Y :=
                (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                        (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                    (P := P) (Q := ⊤) (W := ⊤).left)
                (CommaMorphism.left
                  (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                    ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
                (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
            ⋯ ⋯)
          trivial trivial))

Tactic:
refine 𝒱.hom_ext _ _ fun i ↦ ?_

Premises used: AlgebraicGeometry.Scheme.Cover.hom_ext
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
m : d.gluedCocone (P := P).pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j
i : 𝒱.I₀
⊢ Iso.inv (d.pullbackGluedIso (P := P) i) ≫ CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m =
    (d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)

Tactic:
(d.isColimit i).hom_ext fun a ↦ ?_

Premises used: CategoryTheory.Limits.IsColimit.hom_ext, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.isColimit
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
m : d.gluedCocone (P := P).pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j
i : 𝒱.I₀
refine :
  ∀ (a : J),
    (d.cocone (P := P) i).ι.app a ≫
        Iso.inv (d.pullbackGluedIso (P := P) i) ≫
          CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m =
      (d.cocone (P := P) i).ι.app a ≫
        (d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)
⊢ Iso.inv (d.pullbackGluedIso (P := P) i) ≫ CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m =
    (d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)

Tactic:
refine (d.isColimit i).hom_ext fun a ↦ ?_

Premises used: CategoryTheory.Limits.IsColimit.hom_ext, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.isColimit
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
m : d.gluedCocone (P := P).pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j
i : 𝒱.I₀
a : J
rw :
  (d.cocone (P := P) i).ι.app a ≫
      Iso.inv (d.pullbackGluedIso (P := P) i) ≫
        CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m =
    NatTrans.app (Cocone.ι (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)) a
⊢ (d.cocone (P := P) i).ι.app a ≫
      Iso.inv (d.pullbackGluedIso (P := P) i) ≫
        CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m =
    (d.cocone (P := P) i).ι.app a ≫
      (d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)

Tactic:
rw [IsColimit.fac]

Premises used: CategoryTheory.Limits.IsColimit.fac
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
m : d.gluedCocone (P := P).pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j
i : 𝒱.I₀
a : J
h :
  ((d.cocone (P := P) i).ι.app a ≫
            Iso.inv (d.pullbackGluedIso (P := P) i) ≫
              CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m).toCommaMorphism
        (P := P) (Q := ⊤) (W := ⊤).left =
    CommaMorphism.left
      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (NatTrans.app (Cocone.ι (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)) a))
⊢ (d.cocone (P := P) i).ι.app a ≫
      Iso.inv (d.pullbackGluedIso (P := P) i) ≫
        CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m =
    NatTrans.app (Cocone.ι (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)) a

Tactic:
ext