Deitmar's versus Toen-Vaquie's schemes over F_1

We show the equivalence between Deitmar’s and Toen-Vaquie’s notions of schemes over F_1 (the ‘field with one element’), establishing a symmetry with the classical case of schemes, seen either as spaces with a structure sheaf, or functors of points. In proving so, we also conclude some new basic results on commutative algebra of monoids.