  Given two random variables $X$ and $Y$, stochastic monotonicity describes a
monotone influence of $X$ on $Y$. We prove two different characterizations of
stochastically monotone $2$-copulas using the isomorphism between $2$-copulas
and Markov operators. The first approach establishes a one-to-one
correspondence between stochastically monotone copulas and
monotonicity-preserving Markov operators. The second approach characterizes
stochastically monotone copulas by their monotonicity property with respect to
the Markov product. Applying the latter result, we identify all idempotent
stochastically monotone copulas as ordinal sums of the independence copula
$\Pi$.
