I'm trying to read J.S. Milne's notes on Algebraic Groups. $X$ is a functor from the category $\mathscr C$ of $k$-algebras of the form $k[X_1, ... , X_n]/\mathfrak a, \mathfrak a$ an ideal of $k[X_1, ... , X_n]$, to the category of sets, and $U$ is a subfunctor of $X$. If $A$ is an object of $\mathscr C$, let $h^A$ be the functor $\textrm{Hom}(A,-)$. Milne writes "A subfunctor $U$ of $X$ is open if, for all natural transformations $\phi: h^A \rightarrow X$, the subfunctor $\phi^{-1}(U)$ of $h^A$ is defined by an open subscheme of $\textrm{Spm(A)}$." Here $\textrm{Spm(A)}$ is the space of maximal ideals of $A$ together with a certain sheaf of $k$-algebras.
I don't get what it means for a subfunctor $G$ of $h^A$ to be "defined by an open subscheme." Does anyone know?
If I had to guess (now entering make stuff up mode), it would be something like: there exists an open subset $W$ of $\textrm{Spm A}$, such that for all objects $B$ of $\mathscr C$, $G(B)$ consists of all homomorphisms in $\textrm{Hom(A,B)}$ which correspond to morphisms in $\textrm{Hom}(\textrm{Spm} B, \textrm{Spm} A)$ for which the image of the continuous map on the underlying spaces lies in $W$.