I'm confused on the definition of an "$F$-morphism" of $F$-varieties. The textbook is Springer, Linear Algebraic Groups. Let $k$ be an algebraically closed field, and $F$ a subfield of $k$. The notions of an affine $F$-variety, and of a morphism of affine $F$-varieties, are clear to me.
An affine $F$-algebra is a geometrically reduced, finitely generated $F$-algebra $A_0$. One tensors with $k$ over $F$ to get a reduced, finitely generated $k$-algebra $A = k \otimes_F A_0$, and then defines on $\textrm{Spm } A$ a subtopology of '$F$-open sets' satisfying certain properties, and calls $\textrm{Spm } A$ together with the $F$-open sets an affine $F$-variety.
One then defines a morphism of affine $F$-varieties in such a way that the category of affine $F$-algebras is antiequivalent to that of affine $F$-varieties.
Next one defines an arbitrary $F$-variety to be an algebraic variety over $k$ with some extra properties: 
What I don't understand is the definition of a morphism between $F$-varieties. If I'm not mistaken, he is saying that in order that $\phi: X \rightarrow Y$ be a morphism of $F$-varieties, it must be the case that for any $F$-open set $V$ in $Y$, the preimage $\phi^{-1}V$ must be $F$-open in $X$, and the restriction of $\phi$ to $\phi^{-1}V$ must be a morphism of $F$-varieties $\phi^{-1}V \rightarrow V$.
As the definition is, it seems to be circular: what does it mean for $\phi^{-1}V \rightarrow V$ to be a morphism?
Since we do have a well defined notion of a morphism of affine $F$-varieties, my naive guess as to how to fix the definition would be:
If $\phi: X \rightarrow Y$ is a morphism of varieties between $F$-varieties $X, Y$, we say that $\phi$ is a morphism of $F$-varieties if the preimage of any $F$-open subset of $Y$ is $F$-open in $X$, and for any $F$-open affine set $V$, and any affine $F$-open cover $U_i$ of $\phi^{-1}V$, the restriction of $\phi$ to $U_i$ is a morphism of affine $F$-varieties $U_i \rightarrow V$.
