-- 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
a : J
i : 𝒰.I₀
generalize_proofs :
  ∀ (pf : IsMultiplicative.{u, u + 1} (C := Scheme) ⊤) (pf_1 : IsMultiplicative (C := Discrete PUnit.{1}) ⊤)
    (pf_2 : HasPullback (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i))
    (pf_3 : (precoverage IsOpenImmersion).IsStableUnderBaseChange)
    (pf_4 : ∀ (i : 𝒰.I₀), HasPullback (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i))
    (pf_5 : ∀ (i : 𝒰.I₀), HasPullback (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i))
    (pf_6 :
      ∀ (i : (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).I₀),
        P.HasPullbacksAlong (𝒰.f i))
    (pf_7 :
      ∀ (i : (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).I₀),
        P.IsStableUnderBaseChangeAlong (𝒰.f i))
    (pf_8 : IsStableUnderBaseChange.{u, u + 1} (C := Scheme) ⊤)
    (pf_9 : HasColimit d.relativeGluingData (P := P).functor)
    (pf_10 :
      ∀ {j : (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).I₀},
        P.HasPullbacksAlong (𝒰.f j))
    (pf_11 :
      ∀ {j : (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).I₀},
        P.IsStableUnderBaseChangeAlong (𝒰.f j))
    (h1 :
      ∀ {i j : (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).I₀}
        (hij : i ⟶ j),
        Eq (α :=
          (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).X i ⟶
            Over.left
              ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P)
                (Q := ⊤) (W := ⊤)))
          (Cover.trans (P := IsOpenImmersion)
              (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰) hij ≫
            ((d.cocone (P := P) j).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
              colimit.ι d.relativeGluingData (P := P).functor j)
          (((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
            colimit.ι d.relativeGluingData (P := P).functor i))
    (h2 :
      ∀ (i : (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).I₀),
        (((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
              colimit.ι d.relativeGluingData (P := P).functor i) ≫
            Over.hom
              ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P)
                (Q := ⊤) (W := ⊤)) =
          (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).f i ≫
            Over.hom ((D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤)))
    (pf_12 : P.HasPullbacksAlong (𝒰.f i)) (pf_13 : P.IsStableUnderBaseChangeAlong (𝒰.f i)),
    pullback.fst (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) ≫
        (OpenCover.glueMorphismsOverOfLocallyDirected
            (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰)
            (fun (i : (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).I₀) ↦
              ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                colimit.ι d.relativeGluingData (P := P).functor i)
            (@h1) h2).left =
      ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
        colimit.ι d.relativeGluingData (P := P).functor i
⊢ Eq (α :=
    pullback (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) ⟶
      (Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q := ⊤)
          (W := ⊤).left)
    (pullback.fst (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) ≫
      CommaMorphism.left
        (OpenCover.glueMorphismsOverOfLocallyDirected
          (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰)
          (fun (i : (Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰).I₀) ↦
            ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
              colimit.ι d.relativeGluingData (P := P).functor i)
          ⋯ ⋯))
    (((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
      colimit.ι d.relativeGluingData (P := P).functor i)

Tactic:
generalize_proofs _ _ _ _ _ _ _ _ _ _ _ _ h1 h2