-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits (hypothesis)
Goal:
ι : Type v
inst : Small.{u, v} ι
F : CategoryTheory.Functor.{v, u, v, u + 1} (Discrete ι) LocallyRingedSpace
s : Cocone F
i : Discrete ι
y : (↑(↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace : TopCat) : Type u)
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ IsLocalHom (R :=
    (↑(s.pt.presheaf.stalk
          (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) (X := (↑(coproductCofan F).pt.toPresheafedSpace : TopCat)) (Y :=
                  (↑s.pt.toPresheafedSpace : TopCat))
                  (PresheafedSpace.Hom.base
                      (InducedCategory.Hom.hom
                        (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))) :
                    (↑(coproductCofan F).pt.toPresheafedSpace : TopCat) ⟶ (↑s.pt.toPresheafedSpace : TopCat)) :
                C((↑(↑(coproductCofan F).pt.toPresheafedSpace : TopCat) : Type u),
                  (↑(↑s.pt.toPresheafedSpace : TopCat) : Type u))) :
              (a : (↑(↑(coproductCofan F).pt.toPresheafedSpace : TopCat) : Type u)) →
                (↑(↑s.pt.toPresheafedSpace : TopCat) : Type u))
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                    TopCat.carrier) (Y := (↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat))
                    (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (colimit.ι (F ⋙ forgetToSheafedSpace) i))) :
                  (a : (↑(↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace : TopCat) : Type u)) →
                    (↑(↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat) : Type u))
                y :
              (↑(↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat) : Type u)))) :
      Type u))
    (S :=
    (↑((coproductCofan F).pt.presheaf.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                (Y := (↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat))
                (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (colimit.ι (F ⋙ forgetToSheafedSpace) i))) :
              (a : (↑(↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace : TopCat) : Type u)) →
                (↑(↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat) : Type u))
            y)) :
      Type u))
    (F :=
    (↑(s.pt.presheaf.stalk
            (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                    TopCat.carrier) (X := (↑(coproductCofan F).pt.toPresheafedSpace : TopCat)) (Y :=
                    (↑s.pt.toPresheafedSpace : TopCat))
                    (PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom
                          (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))) :
                      (↑(coproductCofan F).pt.toPresheafedSpace : TopCat) ⟶ (↑s.pt.toPresheafedSpace : TopCat)) :
                  C((↑(↑(coproductCofan F).pt.toPresheafedSpace : TopCat) : Type u),
                    (↑(↑s.pt.toPresheafedSpace : TopCat) : Type u))) :
                (a : (↑(↑(coproductCofan F).pt.toPresheafedSpace : TopCat) : Type u)) →
                  (↑(↑s.pt.toPresheafedSpace : TopCat) : Type u))
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                      TopCat.carrier) (Y := (↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat))
                      (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (colimit.ι (F ⋙ forgetToSheafedSpace) i))) :
                    (a : (↑(↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace : TopCat) : Type u)) →
                      (↑(↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat) : Type u))
                  y :
                (↑(↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat) : Type u)))) :
        Type u) →+*
      (↑((coproductCofan F).pt.presheaf.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) (Y := (↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat))
                  (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (colimit.ι (F ⋙ forgetToSheafedSpace) i))) :
                (a : (↑(↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace : TopCat) : Type u)) →
                  (↑(↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat) : Type u))
              y)) :
        Type u))
    (CommRingCat.Hom.hom
      (PresheafedSpace.Hom.stalkMap
        (InducedCategory.Hom.hom (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              (Y := (↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat))
              (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (colimit.ι (F ⋙ forgetToSheafedSpace) i))) :
            (a : (↑(↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace : TopCat) : Type u)) →
              (↑(↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat) : Type u))
          y)))
have this₁ : Eq (α :=
  (forgetToSheafedSpace.mapCocone s).pt.presheaf.stalk
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            (Y := (↑(forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace : TopCat))
            (PresheafedSpace.Hom.base
              (InducedCategory.Hom.hom
                (CategoryStruct.comp (Y := colimit (F ⋙ forgetToSheafedSpace)) (colimit.ι (F ⋙ forgetToSheafedSpace) i)
                  (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))))) :
          (a : (↑(↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace : TopCat) : Type u)) →
            (↑(↑(forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace : TopCat) : Type u))
        y) ⟶
    (colimit (F ⋙ forgetToSheafedSpace)).presheaf.stalk
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            (Y := (↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat))
            (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (colimit.ι (F ⋙ forgetToSheafedSpace) i))) :
          (a : (↑(↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace : TopCat) : Type u)) →
            (↑(↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat) : Type u))
        y))
  ((eqToHom sorry ≫ PresheafedSpace.Hom.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i).hom y) ≫
    inv (PresheafedSpace.Hom.stalkMap (InducedCategory.Hom.hom (colimit.ι (F ⋙ forgetToSheafedSpace) i)) y))
  (PresheafedSpace.Hom.stalkMap
    (InducedCategory.Hom.hom (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) (Y :=
          (↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat))
          (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (colimit.ι (F ⋙ forgetToSheafedSpace) i))) :
        (a : (↑(↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace : TopCat) : Type u)) →
          (↑(↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace : TopCat) : Type u))
      y))
Tactic: simp only [← IsIso.comp_inv_eq,
          ← InducedCategory.comp_hom,
          PresheafedSpace.stalkMap.congr_hom _ _
            (congr_arg (InducedCategory.Hom.hom) (colimit.ι_desc
              (forgetToSheafedSpace.mapCocone s) i))] at this
Cleared hypotheses: this