∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : UniformSpace β] {𝔖 : Set (Set α)} [inst_1 : UniformSpace γ]
  {f : γ → β},
  IsUniformInducing f → IsUniformInducing (⇑(UniformOnFun.ofFun 𝔖) ∘ (fun x => f ∘ x) ∘ ⇑(UniformOnFun.toFun 𝔖))