-- 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))
i j : 𝒰.I₀
hij : i ⟶ j
iso1 : (d.cocone (P := P) i).pt ≅
  Cocone.pt (F :=
    (D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) ⋙
      MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
    (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
      (d.cocone (P := P) j)) :=
  (d.isColimit (P := P) i).coconePointsIsoOfNatIso
    (isColimitOfPreserves (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
      (d.isColimit (P := P) j))
    (D.isoWhiskerLeft
        (MorphismProperty.Over.pullbackComp (P := P) (Q := ⊤) (Cover.trans (P := IsOpenImmersion) 𝒰 hij) (𝒰.f j) (𝒰.f i)
          ⋯) ≪≫
      (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))).symm)
heq :
  Eq (α :=
    Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt ⟶
      Functor.obj (𝟭 (P.Over ⊤ (𝒰.X j))) (d.cocone (P := P) j).pt)
    (CategoryTheory.Functor.map (MorphismProperty.Over.map (P := P) ⊤ ⋯) iso1.hom ≫
        NatTrans.app
          (Adjunction.counit
            (MorphismProperty.Over.mapPullbackAdj P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij) ⋯ trivial))
          (d.cocone (P := P) j).pt :
      Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt ⟶
        Functor.obj (𝟭 (P.Over ⊤ (𝒰.X j))) (d.cocone (P := P) j).pt)
    (d.transitionMap (P := P) hij :
      Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt ⟶ (d.cocone (P := P) j).pt)
iso2 : (d.cocone (P := P) i).pt.toComma (P := P) (Q := ⊤) (W := ⊤).left ≅
  pullback (d.cocone (P := P) j).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (Cover.trans (P := IsOpenImmersion) 𝒰 hij) :=
  Functor.mapIso (MorphismProperty.Over.forget P ⊤ (𝒰.X i) ⋙ CategoryTheory.Over.forget (𝒰.X i)) iso1
refine_1 :
  pullback.fst (d.cocone (P := P) j).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom
        (Cover.trans (P := IsOpenImmersion) 𝒰 hij) ≫
      Iso.hom (Iso.refl (Functor.obj (𝟭 Scheme) (d.cocone (P := P) j).pt.toComma (P := P) (Q := ⊤) (W := ⊤).left)) =
    iso2.symm.hom ≫
      CommaMorphism.left (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤) (d.transitionMap (P := P) hij))
refine_2 :
  pullback.snd (d.cocone (P := P) j).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom
        (Cover.trans (P := IsOpenImmersion) 𝒰 hij) ≫
      Iso.hom (Iso.refl (𝒰.X i)) =
    iso2.symm.hom ≫ (d.cocone (P := P) i).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom
⊢ IsPullback (CommaMorphism.left (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤) (d.transitionMap (P := P) hij)))
    (d.cocone (P := P) i).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom
    (d.cocone (P := P) j).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (Cover.trans (P := IsOpenImmersion) 𝒰 hij)

Tactic:
refine .of_iso (IsPullback.of_hasPullback _ _) iso2.symm (.refl _) (.refl _) (.refl _) ?_ ?_
      (by simp) (by simp)