  In the case of rational Cherednik algebras associated with cyclic groups, we
give an alternative proof that the projective object $P_{\text{KZ}}$
representing the KZ-functor is isomorphic to the $\Delta$-module associated
with the coinvariant algebra for a subset of parameter values from which all
parameter values can be obtained by integral translations. We also specify the
exact parameter values for which this isomorphism occurs. Furthermore, we
determine the action of the cyclotomic Hecke algebra on $P_{\text{KZ}}$ for
these parameter values, thereby giving a complete algebraic description of the
KZ-functor in this case.
