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 \prec_X y \quad \implies \quad f(y) \prec_Y f(x) \right) \end{equation}