  In the recent paper [Stud. App. Math. 147 (2021) 752], squared eigenfunction
symmetry constraint of the differential-difference modified
Kadomtsev-Petviashvili (D$\Delta$mKP) hierarchy converts the D$\Delta$mKP
system to the relativistic Toda spectral problem and its hierarchy. In this
paper we introduce a new formulation of independent variables in the squared
eigenfunction symmetry constraint, under which the D$\Delta$mKP system gives
rise to the discrete spectral problem and a hierarchy of the
differential-difference derivative nonlinear Schr\"odinger equation of the
Chen-Lee-Liu type. In addition, by introducing nonisospectral flows, two sets
of symmetries of the D$\Delta$mKP hierarchy and their algebraic structure are
obtained. We then present a unified continuum limit scheme, by which we achieve
the correspondence of the mKP and the D$\Delta$mKP hierarchies and their
integrable structures.
