We prove that the two constructions of $\Lambda $-maps given by D. Grayson and the author by means of certain (multi) simplicial sets provide the same definition for $\lambda $-operations on higher $K$-theory of the underlying exact category with operations. The equivalence is proved on the simplicial level. Our main technical result is that, under certain assumptions, the multidimensional mapping cone construction of a cube of exact categories may be computed as a homotopy fiber of the map of multidimensional $S$-constructions applied to faces of codimension one, which is a generalization of Grayson's Theorem $C$.