Let $P_X = (X, {\preceq_X})$ and $P_Y = (Y, {\preceq_Y})$ each be a D1103: Partially ordered set.
A D18: Map $f : X \to Y$ is monotone from $P_X$ to $P_Y$ if and only if at least one of the following statements is true

(1)	$\forall \, x, y \in X \, ((x, y) \in {\preceq_X} \quad \Rightarrow \quad (f(x), f(y)) \in {\preceq_Y})$ (D427: Isotone map)
(2)	$\forall \, x, y \in X \, ((x, y) \in {\preceq_X} \quad \Rightarrow \quad (f(y), f(x)) \in {\preceq_Y})$ (D428: Antitone map)

Let $P_X = (X, {\preceq_X})$ and $P_Y = (Y, {\preceq_Y})$ each be a D1103: Partially ordered set.
A D18: Map $f : X \to Y$ is monotone from $P_X$ to $P_Y$ if and only if at least one of the following statements is true

(1)	$\forall \, x, y \in X \, (x \preceq_X y \quad \Rightarrow \quad f(x) \preceq_Y f(y))$ (D427: Isotone map)
(2)	$\forall \, x, y \in X \, (x \preceq_X y \quad \Rightarrow \quad f(y) \preceq_Y f(x))$ (D428: Antitone map)