∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} {N : Type u_2} [inst_1 : CommMonoid N] {P : Type u_3}
  [inst_2 : CommMonoid P] (f : S.LocalizationMap N) {g : M →* P} {T : Submonoid P} (hy : ∀ (y : ↥S), g ↑y ∈ T)
  {Q : Type u_4} [inst_3 : CommMonoid Q] {k : T.LocalizationMap Q} (x : M) (y : ↥S),
  (f.map hy k) (f.mk' x y) = k.mk' (g x) ⟨g ↑y, ⋯⟩