∀ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
  [inst_2 : TopologicalSpace Z] {g : Y → Z} {f : X → Y} {s : Set X} {t : Set Y},
  IsLocalHomeomorphOn g t → IsLocalHomeomorphOn f s → Set.MapsTo f s t → IsLocalHomeomorphOn (g ∘ f) s