Morphism of integral $k$-schemes of finite type and closed points

Question

I'm trying to understand the proof that the categories of prevarieties is equivalent to the category of integral $k$-schemes of finite type, as done in Görtz-Wedhorn's book. In this proof, one considers the functor that associates to a integral $k$-scheme of finite type the space $X(k)$ of closed points in $X$. If $f: X \to Y$ is a morphism of integral $k$-schemes of finite type, then one can define $f(k)$ - the image of $f$ under the above defined functor - as simply the restriction of $f$ to $X(k)$ (as a continuous map, I mean). I'm trying to show that this definition makes sense, i.e., that $f(X(k)) \subseteq Y(k)$, but I'm kind of stuck. Can anyone help me?