  In this note, we introduce a natural notion of intrinsic Hopf-Lax semigroup
in the context of the so-called intrinsically Lipschitz sections. The main aims
are to prove the link between the intrinsic Hopf-Lax semigroup and the
intrinsic slope and to show that the intrinsic Hopf-Lax semigroup is a
subsolution of Hamilton-Jacobi type equality.
