Duplicial sets, crossed $G$-sets and descent categories

The goal of this note is to show that the left $\chi$-coalgebra, which is an additional structure on one of the coefficients used in the construction of the cyclic operator for the cyclic sets that generalises the twisted nerve of a group by Loday is an instance of a general theory of left $\chi$-coalgebras. It is also shown that in the general case, the resulting cyclic operator requires that the other coefficient needs to be equipped with a crossed $G$-sets structure.