-- 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
as : ι
this_1 :
  Eq (α := (F ⋙ forget C).obj { as := as : Discrete ι } ⟶ colimit (F ⋙ forget C))
    (colimit.ι (F ⋙ forget C) { as := as })
    (CategoryStruct.comp (Y := colimit (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)))
      (NatTrans.app (Iso.hom Discrete.natIsoFunctor) { as := as : Discrete ι } ≫
        colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := as })
      (Iso.inv (HasColimit.isoOfNatIso Discrete.natIsoFunctor)))
this :
  Eq (α :=
    Functor.obj (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk))
        { as := { as := as : Discrete ι }.as : Discrete ι } ⟶
      colimit (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)))
    (colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := { as := as }.as })
    (TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) { as := as }.as ≫
      (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).inv)
mk :
  IsOpenEmbedding (X :=
    (↑(Functor.obj (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := as : Discrete ι }) : Type v)) (Y :=
    (i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
          (TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) as)) :
      (a : (↑((Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)).obj { as := as }) : Type v)) →
        (i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))
⊢ IsOpenEmbedding (X := (↑(↑(F.obj { as := as : Discrete ι }).toPresheafedSpace : TopCat) : Type v)) (Y :=
    (↑(↑(colimit F).toPresheafedSpace : TopCat) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(F.obj { as := as : Discrete ι }).toPresheafedSpace : TopCat)) (Y :=
            (↑(colimit F).toPresheafedSpace : TopCat))
            (CategoryStruct.comp (Y := colimit (F ⋙ forget C))
                (((NatTrans.app (Iso.hom Discrete.natIsoFunctor) { as := as : Discrete ι } ≫
                        TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) { as := as : Discrete ι }.as) ≫
                      (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).inv) ≫
                    Iso.inv (HasColimit.isoOfNatIso Discrete.natIsoFunctor) :
                  (F ⋙ forget C).obj { as := as : Discrete ι } ⟶ colimit (F ⋙ forget C))
                (Iso.inv (preservesColimitIso (forget C) F) :
                  colimit (F ⋙ forget C) ⟶ Functor.obj (forget C) (colimit F)) :
              (F ⋙ forget C).obj { as := as : Discrete ι } ⟶ Functor.obj (forget C) (colimit F)) :
          C((↑(↑(F.obj { as := as : Discrete ι }).toPresheafedSpace : TopCat) : Type v),
            (↑(↑(colimit F).toPresheafedSpace : TopCat) : Type v))) :
      (a : (↑(↑(F.obj { as := as }).toPresheafedSpace : TopCat) : Type v)) →
        (↑(↑(colimit F).toPresheafedSpace : TopCat) : Type v))

Tactic:
simp_rw [TopCat.isOpenEmbedding_iff_comp_isIso, (TopCat.isOpenEmbedding_iff_isIso_comp)]

Premises used: TopCat.isOpenEmbedding_iff_comp_isIso, TopCat.isOpenEmbedding_iff_isIso_comp