-- Module: Mathlib.Geometry.RingedSpace.OpenImmersion
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasLimits.{v, u} C
ι : Type v
F : Discrete ι ⥤ SheafedSpace.{u, v, v} C
inst_2 : HasColimit F
i : Discrete ι
inst_3 : HasStrictTerminalObjects.{v, u} C
U : Opens (↑(↑(F.obj i).toPresheafedSpace : TopCat) : Type v)
h₁ :
  Eq (α := (F ⋙ forgetToPresheafedSpace).obj i ⟶ Functor.obj forgetToPresheafedSpace (colimit F))
    (CategoryStruct.comp (Y := colimit (F ⋙ forgetToPresheafedSpace))
      (colimit.ι (F ⋙ forgetToPresheafedSpace) i :
        (F ⋙ forgetToPresheafedSpace).obj i ⟶ colimit (F ⋙ forgetToPresheafedSpace))
      (Iso.inv (preservesColimitIso forgetToPresheafedSpace F) :
        colimit (F ⋙ forgetToPresheafedSpace) ⟶ Functor.obj forgetToPresheafedSpace (colimit F)))
    (CategoryTheory.Functor.map forgetToPresheafedSpace (colimit.ι F i))
h₂ :
  Eq (α := F.obj i ⟶ colimit F) (colimit.ι F i)
    {
      hom :=
        CategoryStruct.comp (Y := colimit (F ⋙ forgetToPresheafedSpace))
          (colimit.ι (F ⋙ forgetToPresheafedSpace) i :
            (F ⋙ forgetToPresheafedSpace).obj i ⟶ colimit (F ⋙ forgetToPresheafedSpace))
          (Iso.inv (preservesColimitIso forgetToPresheafedSpace F) :
            colimit (F ⋙ forgetToPresheafedSpace) ⟶ Functor.obj forgetToPresheafedSpace (colimit F)) }
H :
  IsOpenEmbedding (X := (↑(↑((F ⋙ forgetToPresheafedSpace).obj i) : TopCat) : Type v)) (Y :=
    (↑(↑(Functor.obj forgetToPresheafedSpace (colimit F)) : TopCat) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
          (Y := (↑(Functor.obj forgetToPresheafedSpace (colimit F)) : TopCat))
          (PresheafedSpace.Hom.base
            (CategoryStruct.comp (Y := colimit (F ⋙ forgetToPresheafedSpace))
              (colimit.ι (F ⋙ forgetToPresheafedSpace) i :
                (F ⋙ forgetToPresheafedSpace).obj i ⟶ colimit (F ⋙ forgetToPresheafedSpace))
              (Iso.inv (preservesColimitIso forgetToPresheafedSpace F) :
                colimit (F ⋙ forgetToPresheafedSpace) ⟶ Functor.obj forgetToPresheafedSpace (colimit F))))) :
      (a : (↑(↑((F ⋙ forgetToPresheafedSpace).obj i) : TopCat) : Type v)) →
        (↑(↑(forgetToPresheafedSpace.obj (colimit F)) : TopCat) : Type v))
this :
  IsIso (C := TopCat.Presheaf.{v, v, u} C (↑(Functor.obj forgetToPresheafedSpace (colimit F)) : TopCat)) (Y :=
    Functor.obj
      (TopCat.Presheaf.pushforward C
          (PresheafedSpace.Hom.base (Iso.inv (preservesColimitIso forgetToPresheafedSpace F))) :
        TopCat.Presheaf.{v, v, u} C (↑(colimit (F ⋙ forgetToPresheafedSpace)) : TopCat) ⥤
          TopCat.Presheaf.{v, v, u} C (↑(Functor.obj forgetToPresheafedSpace (colimit F)) : TopCat))
      (colimit (F ⋙ forgetToPresheafedSpace)).presheaf)
    (PresheafedSpace.Hom.c (Iso.inv (preservesColimitIso forgetToPresheafedSpace F)) :
      (Functor.obj forgetToPresheafedSpace (colimit F)).presheaf ⟶
        Functor.obj
          (TopCat.Presheaf.pushforward C
              (PresheafedSpace.Hom.base (Iso.inv (preservesColimitIso forgetToPresheafedSpace F))) :
            TopCat.Presheaf.{v, v, u} C (↑(colimit (F ⋙ forgetToPresheafedSpace)) : TopCat) ⥤
              TopCat.Presheaf.{v, v, u} C (↑(Functor.obj forgetToPresheafedSpace (colimit F)) : TopCat))
          (colimit (F ⋙ forgetToPresheafedSpace)).presheaf)
j : Discrete ι
hj : op j ≠ op i
H₁ :
  IsTerminal.{v, u}
    ((F.obj j).presheaf.obj
      (op
        (Functor.obj (C := Opens (↑(↑(colimit (F ⋙ forgetToPresheafedSpace)) : TopCat) : Type v))
          (Opens.map (PresheafedSpace.Hom.base (colimit.ι (F ⋙ forgetToPresheafedSpace) j)) :
            Opens (↑(↑(colimit (F ⋙ forgetToPresheafedSpace)) : TopCat) : Type v) ⥤
              Opens (↑(↑(F.obj j).toPresheafedSpace : TopCat) : Type v))
          (Functor.obj (Opens.map (PresheafedSpace.Hom.base (Iso.inv (preservesColimitIso forgetToPresheafedSpace F))))
              (H.functor.obj U) :
            Opens (↑(↑(colimit (F ⋙ forgetToPresheafedSpace)) : TopCat) : Type v)))))
⊢ IsTerminal.{v, u}
    ((PresheafedSpace.componentwiseDiagram (F ⋙ forgetToPresheafedSpace)
          (Functor.obj
              (Opens.map (PresheafedSpace.Hom.base (Iso.inv (preservesColimitIso forgetToPresheafedSpace F))) :
                Opens (↑(↑(Functor.obj forgetToPresheafedSpace (colimit F)) : TopCat) : Type v) ⥤
                  Opens (↑(↑(colimit (F ⋙ forgetToPresheafedSpace)) : TopCat) : Type v))
              (H.functor.obj U) :
            Opens (↑(↑(colimit (F ⋙ forgetToPresheafedSpace)) : TopCat) : Type v))).obj
      (op j))

Tactic:
dsimp