∀ {M : Type u_1} {N : Type u_2} [inst : Monoid M] [inst_1 : Monoid N] (u : (M × N)ˣ),
  (fun u => { val := (↑u.1, ↑u.2), inv := (↑u.1⁻¹, ↑u.2⁻¹), val_inv := ⋯, inv_val := ⋯ })
      (((Units.map (MonoidHom.fst M N)).prod (Units.map (MonoidHom.snd M N))) u) =
    u