-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (hypothesis)
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)
⊢ 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)
have heq : Eq (α :=
  Functor.obj (MorphismProperty.Over.map (P := P) ⊤ sorry) (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) ⊤ sorry) iso1.hom ≫
      NatTrans.app
        (Adjunction.counit
          (MorphismProperty.Over.mapPullbackAdj P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij) sorry trivial))
        (d.cocone (P := P) j).pt :
    Functor.obj (MorphismProperty.Over.map (P := P) ⊤ sorry) (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) ⊤ sorry) (d.cocone (P := P) i).pt ⟶ (d.cocone (P := P) j).pt)
Tactic: have heq : (Over.map _ (d.prop_trans hij)).map iso1.hom ≫
        (Over.mapPullbackAdj _ _ _ (d.prop_trans hij) trivial).counit.app _ =
      d.transitionMap hij := by
    apply (isColimitOfPreserves (Over.map _ (d.prop_trans hij)) (d.isColimit i)).hom_ext
    intro a
    dsimp only [Functor.comp_obj, Functor.id_obj, Functor.mapCocone_pt, Functor.const_obj_obj,
      Functor.mapCocone_ι_app, Iso.trans_hom, Functor.isoWhiskerLeft_hom, Iso.symm_hom]
    rw [IsColimit.coconePointsIsoOfNatIso_hom, ← Functor.map_comp_assoc, IsColimit.ι_map,
      Functor.mapCocone_ι_app, Functor.map_comp, Category.assoc,
      Adjunction.counit_naturality, cocone_ι_transitionMap]
    ext
    dsimp
    rw [Category.comp_id, ← Category.assoc]
    congr 1
    apply pullback.hom_ext <;> simp