Let $P_X = (X, {\prec_X})$ and $P_Y = (Y, {\prec_Y})$ each be a D1776: Strict partially ordered set.
A D18: Map $f : X \to Y$ is strictly antitone from $P_X$ to $P_Y$ if and only if \begin{equation} \forall \, x, y \in X \left( (x, y) \in {\prec_X} \quad \implies \quad (f(y), f(x)) \in {\prec_Y} \right) \end{equation}