On doubly minimal systems and a question regarding product recurrence

We show that a doubly minimal system $X$ has the property that for every minimal system $Y$ the orbit closure of any pair $(y,x) \in Y \times X$ is either $Y \times X$ or it has the form $\Gamma_\pi = \{(\pi(x),x) : x \in X\}$ for some factor map $\pi: X \to Y$. As a corollary we resolve a problem of Haddad and Ott from 2008 regarding product recurrence.