instance sigma_ι_isOpenImmersion [HasStrictTerminalObjects C] :
    SheafedSpace.IsOpenImmersion (colimit.ι F i) where
  base_open := sigma_ι_openEmbedding F i
  c_iso U := by
    have e : colimit.ι F i = _ :=
      (ι_preservesColimitsIso_inv SheafedSpace.forgetToPresheafedSpace F i).symm
    have H :
      OpenEmbedding
        (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) i ≫
            (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv).base :=
      e ▸ sigma_ι_openEmbedding F i
    suffices IsIso <| (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) i ≫
        (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv).c.app <|
      op (H.isOpenMap.functor.obj U) by
      -- Porting note : just `convert` is very slow, so helps it a bit
      convert this using 2 <;> congr
    rw [PresheafedSpace.comp_c_app,
      ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
    -- Porting note : this instance created manually to make the `inferInstance` below work
    have inst1 : IsIso (preservesColimitIso forgetToPresheafedSpace F).inv.c :=
      PresheafedSpace.c_isIso_of_iso _
    rsuffices : IsIso
        (limit.π
          (PresheafedSpace.componentwiseDiagram (F ⋙ SheafedSpace.forgetToPresheafedSpace)
            ((Opens.map
                  (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
              (unop <| op <| H.isOpenMap.functor.obj U)))
          (op i))
    · 

Proof state:
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝¹ : HasColimit F
i : Discrete ι
inst✝ : HasStrictTerminalObjects C
U : Opens ↑↑(F.obj i).toPresheafedSpace
e : colimit.ι F i = colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv
H :
  OpenEmbedding ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv).base
inst1 : IsIso (preservesColimitIso forgetToPresheafedSpace F).inv.c
this✝ :
  IsIso
    (limit.π
      (PresheafedSpace.componentwiseDiagram (F ⋙ forgetToPresheafedSpace)
        ((Opens.map (preservesColimitIso forgetToPresheafedSpace F).inv.base).obj
          (op
              ((IsOpenMap.functor
                    (_ :
                      IsOpenMap
                        ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫
                              (preservesColimitIso forgetToPresheafedSpace F).inv).base)).obj
                U)).unop))
      (op i))
⊢ IsIso
    ((preservesColimitIso forgetToPresheafedSpace F).inv.c.app
        (op
          ((IsOpenMap.functor
                (_ :
                  IsOpenMap
                    ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫
                          (preservesColimitIso forgetToPresheafedSpace F).inv).base)).obj
            U)) ≫
      (PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit (F ⋙ forgetToPresheafedSpace)
            ((Opens.map (preservesColimitIso forgetToPresheafedSpace F).inv.base).obj
              (op
                  ((IsOpenMap.functor
                        (_ :
                          IsOpenMap
                            ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫
                                  (preservesColimitIso forgetToPresheafedSpace F).inv).base)).obj
                    U)).unop)).hom ≫
        limit.π
          (PresheafedSpace.componentwiseDiagram (F ⋙ forgetToPresheafedSpace)
            ((Opens.map (preservesColimitIso forgetToPresheafedSpace F).inv.base).obj
              (op
                  ((IsOpenMap.functor
                        (_ :
                          IsOpenMap
                            ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫
                                  (preservesColimitIso forgetToPresheafedSpace F).inv).base)).obj
                    U)).unop))
          (op i))

Next tactic:
infer_instance