-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
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 : (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
a b : J
f : a ⟶ b
⊢ Eq (α :=
    (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left ⟶
      (Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) b).toComma (P := P) (Q := ⊤)
          (W := ⊤).left)
    (CommaMorphism.left
      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (D.map f ≫
          Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
            (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 b)
              (fun (i : (𝒱 b).I₀) ↦
                ((d.cocone (P := P) i).ι.app b).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                  colimit.ι d.relativeGluingData (P := P).functor i)
              ⋯ ⋯)
            trivial trivial)))
    (CommaMorphism.left
      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
            (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 a)
              (fun (i : (𝒱 a).I₀) ↦
                ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                  colimit.ι d.relativeGluingData (P := P).functor i)
              ⋯ ⋯)
            trivial trivial ≫
          CategoryTheory.Functor.map (Functor.obj (Functor.const J) (d.glued (P := P))) f)))
have refine_2 : ∀ (x : (𝒱 a).I₀),
  Eq (α :=
    (𝒱 a).X x ⟶
      (Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) b).toComma (P := P) (Q := ⊤)
          (W := ⊤).left)
    ((𝒱 a).f x ≫
      CommaMorphism.left
        (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
          (D.map f ≫
            Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
              (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 b)
                (fun (i : (𝒱 b).I₀) ↦
                  ((d.cocone (P := P) i).ι.app b).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                    colimit.ι d.relativeGluingData (P := P).functor i)
                sorry sorry)
              trivial trivial)))
    ((𝒱 a).f x ≫
      CommaMorphism.left
        (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
          (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
              (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 a)
                (fun (i : (𝒱 a).I₀) ↦
                  ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                    colimit.ι d.relativeGluingData (P := P).functor i)
                sorry sorry)
              trivial trivial ≫
            CategoryTheory.Functor.map (Functor.obj (Functor.const J) (d.glued (P := P))) f)))
Tactic: apply (𝒱 a).hom_ext