-- 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
i j : 𝒱.I₀
hij : i ⟶ j
refine_1 :
  Eq (α :=
    (Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt).toComma (P := P) (Q := ⊤) (W :=
          ⊤).left ⟶
      Over.left (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤)))
    (CommaMorphism.left (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤) (d.transitionMap (P := P) hij)) ≫
        CategoryStruct.comp (Y := pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f j))
          (CommaMorphism.left
              (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                ((d.isColimit (P := P) j).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s))) :
            (d.cocone (P := P) j).pt.toComma (P := P) (Q := ⊤) (W := ⊤).left ⟶
              (Functor.obj (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s.pt).toComma (P := P) (Q := ⊤) (W := ⊤).left)
          (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f j) :
            pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f j) ⟶
              s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).left) :
      (Functor.obj (MorphismProperty.Over.map (P := P) ⊤ ⋯) (d.cocone (P := P) i).pt).toComma (P := P) (Q := ⊤) (W :=
            ⊤).left ⟶
        Over.left (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤)))
    (CategoryStruct.comp (Y := pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i))
        (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))) :
          (d.cocone (P := P) i).pt.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) ⟶ s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).left) :
      𝒱.X i ⟶ Over.left (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤)))
⊢ Eq (α := 𝒱.X i ⟶ Over.left (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤)))
    (Cover.trans (P := IsOpenImmersion) 𝒱 hij ≫
      CategoryStruct.comp (Y := pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f j))
        (CommaMorphism.left
            (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
              ((d.isColimit (P := P) j).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s))) :
          (d.cocone (P := P) j).pt.toComma (P := P) (Q := ⊤) (W := ⊤).left ⟶
            (Functor.obj (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s.pt).toComma (P := P) (Q := ⊤) (W := ⊤).left)
        (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f j) :
          pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f j) ⟶ s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).left))
    (CategoryStruct.comp (Y := pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i))
      (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))) :
        (d.cocone (P := P) i).pt.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) ⟶ s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).left))

Tactic:
change (d.transitionMap hij).left ≫ _ ≫ _ = _

Premises used: AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionMap, CategoryTheory.CommaMorphism.left, CategoryTheory.MorphismProperty.Comma.Hom.toCommaMorphism