∀ {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Monoid α] [inst_1 : Monoid β] [inst_2 : FunLike F α β]
  [MonoidHomClass F α β] (f : F) [IsLocalHom f] (a : α), (f a ∈ nonunits β) = (a ∈ nonunits α)