  The Z-property of a linear map with respect to a cone is an extension of the
notion of Z-matrices. In a recent paper of Orlitzky (see Corollary 6.2 in M.
Orlitzky. Positive and $\mathbf{Z}$-operators on closed convex cones, Electron.
J Linear Algebra, 444--458, 2018) the characterisation of cone-complementarity
is given in terms of the dual of the cone of linear maps satisfying the
Z-property. Therefore, it is meaningful to consider the problem of finding the
dual cone of the cone of linear maps which have the Z-property with respect to
a cone. This short note will solve this problem in the particular case when the
Z-property is considered with respect to the Lorentz cone.
