-- 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
m : d.gluedCocone (P := P).pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j
i : 𝒱.I₀
a : J
h :
  ((d.cocone (P := P) i).ι.app a ≫
              Iso.inv (d.pullbackGluedIso (P := P) i) ≫
                CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m).toCommaMorphism
          (P := P) (Q := ⊤) (W := ⊤).left ≫
      pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) =
    CategoryStruct.comp (Y := pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i))
      (CommaMorphism.left
          (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
            (NatTrans.app (Cocone.ι (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)) a)) :
        ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left ⟶
          (Functor.obj (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s.pt).toComma (P := P) (Q := ⊤) (W := ⊤).left)
      (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) :
        pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) ⟶
          Functor.obj (𝟭 Scheme) s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).left)
h₁ :
  ((d.cocone (P := P) i).ι.app a ≫
              Iso.inv (d.pullbackGluedIso (P := P) i) ≫
                CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m).toCommaMorphism
          (P := P) (Q := ⊤) (W := ⊤).left ≫
      pullback.snd s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) =
    CategoryStruct.comp (Y := pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i))
      (CommaMorphism.left
          (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
            (NatTrans.app (Cocone.ι (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)) a)) :
        ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left ⟶
          (Functor.obj (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s.pt).toComma (P := P) (Q := ⊤) (W := ⊤).left)
      (pullback.snd s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) :
        pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) ⟶ 𝒰.X i)
⊢ ((d.cocone (P := P) i).ι.app a ≫
            Iso.inv (d.pullbackGluedIso (P := P) i) ≫
              CategoryTheory.Functor.map (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) m).toCommaMorphism
        (P := P) (Q := ⊤) (W := ⊤).left =
    CommaMorphism.left
      (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
        (NatTrans.app (Cocone.ι (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)) a))

Tactic:
apply pullback.hom_ext

Premises used: CategoryTheory.Limits.pullback.hom_ext