  We prove a functoriality result for the full C*-algebras of right-LCM monoids
with respect to monoid inclusions that are closed under factorization and
preserve orthogonality, and use this to show that if a right-LCM monoid is
amenable in the sense of Nica, then so are its submonoids. As applications, we
complete the classification of Artin monoids with respect to Nica amenability
by showing that only the right-angled ones are amenable in the sense of Nica
and we show that the Nica amenability of a graph product of right-LCM
semigroups is inherited by the factors.
