-- 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 :
  ∀ (i : (𝒲 a).I₀),
    (𝒲 a).f i ≫
        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)) =
      (𝒲 a).f i ≫ (s.ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left
⊢ 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:
refine (𝒲 a).hom_ext _ _ fun i ↦ ?_

Premises used: AlgebraicGeometry.Scheme.Cover.hom_ext