Open subfunctor ($h_T{\times }_{F}H\cong h_U,U\subseteq T\text{ open}$)

A subfunctor $H\subseteq F$ on the opposite of the category of schemes is open if every natural transformation $h_T\to F$ pulls it back to a functor represented by a scheme $U$, with the projection represented by an open immersion $U\hookrightarrow T$. Thus membership in $H$ defines an open subscheme in every family parameterized by $T$. This condition is preserved by further pullbacks.