-- 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
i j : 𝒱.I₀
hij : i ⟶ j
apply :
  ∀ (j_1 : J),
    Eq (α :=
      ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) ⋙ MorphismProperty.Over.map (P := P) ⊤ ⋯).obj j_1 ⟶
        Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s))
      (CategoryStruct.comp (Y :=
        Functor.obj.{v_1, u, u_1, u + 1}
          (Functor.obj (Functor.const J)
            (Cocone.pt (F := (D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) ⋙ MorphismProperty.Over.map (P := P) ⊤ ⋯)
              (Functor.mapCocone (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i))))
          j_1)
        (NatTrans.app (Cocone.ι (Functor.mapCocone (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i))) j_1)
        (d.transitionMap (P := P) hij ≫
          (d.isColimit (P := P) j).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s)))
      (CategoryStruct.comp (Y :=
        Functor.obj.{v_1, u, u_1, u + 1}
          (Functor.obj (Functor.const J)
            (Cocone.pt (F := (D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) ⋙ MorphismProperty.Over.map (P := P) ⊤ ⋯)
              (Functor.mapCocone (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i))))
          j_1)
        (NatTrans.app (Cocone.ι (Functor.mapCocone (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i))) j_1)
        (CategoryStruct.comp (Y :=
          Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯)
            (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
              (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)))
          (CategoryTheory.Functor.map (MorphismProperty.Over.map (P := P) ⊤ ⋯)
            ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)))
          (NatTrans.app
            (Over.pullbackMapHomPullback (P := P) (Q := ⊤) (Cover.trans (P := IsOpenImmersion) 𝒰 hij) ⋯ trivial (𝒰.f j)
              (𝒰.f i) ⋯)
            s.pt)))
⊢ Eq (α :=
    Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt ⟶
      Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
        (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s))
    (d.transitionMap (P := P) hij ≫
        (d.isColimit (P := P) j).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s) :
      Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt ⟶
        Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s))
    (CategoryStruct.comp (Y :=
        Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯)
          (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
            (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)))
        (CategoryTheory.Functor.map (MorphismProperty.Over.map (P := P) ⊤ ⋯)
          ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)))
        (NatTrans.app
          (Over.pullbackMapHomPullback (P := P) (Q := ⊤) (Cover.trans (P := IsOpenImmersion) 𝒰 hij) ⋯ trivial (𝒰.f j)
            (𝒰.f i) ⋯)
          s.pt) :
      Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt ⟶
        Functor.obj (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s.pt)

Tactic:
apply (isColimitOfPreserves (Over.map ⊤ <| d.prop_trans _) (d.isColimit i)).hom_ext

Premises used: CategoryTheory.Limits.IsColimit.hom_ext, CategoryTheory.MorphismProperty.Over.map, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.prop_trans, Top.top, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.isColimit, CategoryTheory.Limits.isColimitOfPreserves