-- 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
i j k : 𝒰.I₀
hij : i ⟶ j
hjk : j ⟶ k
apply :
  ∀ (j_1 : J),
    Eq (α :=
      ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) ⋙ MorphismProperty.Over.map (P := P) ⊤ ⋯).obj j_1 ⟶
        (d.cocone (P := P) k).pt)
      (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 ≫ hjk)))
      (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)
        (NatTrans.app
            (Iso.hom
              (MorphismProperty.Over.mapComp (P := P) ⊤ ⋯ ⋯ (Cover.trans (P := IsOpenImmersion) 𝒰 (hij ≫ hjk)) ⋯))
            (d.cocone (P := P) i).pt ≫
          CategoryStruct.comp (Y := Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) j).pt)
            (CategoryTheory.Functor.map (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.transitionMap (P := P) hij) :
              Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯)
                  (Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt) ⟶
                Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) j).pt)
            (d.transitionMap (P := P) hjk :
              Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) j).pt ⟶
                (d.cocone (P := P) k).pt)))
⊢ Eq (α := Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt ⟶ (d.cocone (P := P) k).pt)
    (d.transitionMap (P := P) (hij ≫ hjk) :
      Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt ⟶ (d.cocone (P := P) k).pt)
    (NatTrans.app
          (Iso.hom (MorphismProperty.Over.mapComp (P := P) ⊤ ⋯ ⋯ (Cover.trans (P := IsOpenImmersion) 𝒰 (hij ≫ hjk)) ⋯))
          (d.cocone (P := P) i).pt ≫
        CategoryStruct.comp (Y := Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) j).pt)
          (CategoryTheory.Functor.map (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.transitionMap (P := P) hij) :
            Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯)
                (Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt) ⟶
              Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) j).pt)
          (d.transitionMap (P := P) hjk :
            Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) j).pt ⟶ (d.cocone (P := P) k).pt) :
      Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt ⟶ (d.cocone (P := P) k).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