Monotonicity is a property of functions that is relative to some ordering relation (e.g., f(x) = x + 42 as a function on natural numbers is monotone with respec

Monotonicity is a property of functions that is relative to some ordering relation (e.g., f(x) = x + 42 as a function on natural numbers is monotone with respect to the natural ordering). In this case, that ordering relation is <:, and covariant functions are precisely those function that are monotone with respect to <:; as are contravariant functions the antitone functions with respect to <:.