Monotone Function

A function f : X → Y : 𝑓 → 𝑋 𝑌 f:X\to Y between two Preorders is monotone if for all x , y : X : 𝑥 𝑦 𝑋 x,y:X , if x ≤ y 𝑥 𝑦 x\leq y , then f ⁢ ( x ) ≤ f ⁢ ( y ) 𝑓 𝑥 𝑓 𝑦 f(x)\leq f(y) . This is a Vertical Decategorification of the notion of Functor.