∀ {M : Type u_1} {N : Type u_2} [inst : AddMonoid M] [inst_1 : AddMonoid N] (u : AddUnits (M × N)),
  (fun u => { val := (↑u.1, ↑u.2), neg := (↑(-u.1), ↑(-u.2)), val_neg := ⋯, neg_val := ⋯ })
      (((AddUnits.map (AddMonoidHom.fst M N)).prod (AddUnits.map (AddMonoidHom.snd M N))) u) =
    u