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 monotone with respect to $P_X$ and $P_Y$ if and only if one of the following statements is true

(1)	$\forall \, x, y \in X \, ((x, y) \in {\prec_X} \quad \Rightarrow \quad (f(x), f(y)) \in {\prec_Y})$ (D430: Strictly isotone map)
(2)	$\forall \, x, y \in X \, ((x, y) \in {\prec_X} \quad \Rightarrow \quad (f(y), f(x)) \in {\prec_Y})$ (D431: Strictly antitone map)

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 monotone with respect to $P_X$ and $P_Y$ if and only if one of the following statements is true

(1)	$\forall \, x, y \in X \, (x \prec_X y \quad \Rightarrow \quad f(x) \prec_Y f(y))$ (D430: Strictly isotone map)
(2)	$\forall \, x, y \in X \, (x \prec_X y \quad \Rightarrow \quad f(y) \prec_Y f(x))$ (D431: Strictly antitone map)