This is definition of scheme functor in Mumford, Oda, Algebraic Geometry II Chpt 1.
Let $F$ be a covariant functor from Ring category to set category s.t. for any ring $R$, $Spec(R)=\cup_iD(f_i)$, $F(R)$ is the equalizer of $\prod_iF(R_{f_i})\to\prod_{ij}F(R_{f_if_j})$. Then $F$ is called sheaf in zariski topology.
Let covariant functor $F:Ring\to Set$ with $x\in F(R)$ called open set if (a) $Hom(Spec(-), Spec(R))\xrightarrow{x_\star} F$ where $x_\star$ is pushforward and both $Hom(Spec(-), Spec(R)), F$ are functors from ring category to sets and (b) For all ring $S,\eta\in F(S)$, then $\eta_\star^{-1}[x_\star(Hom(Spec(-),Spec(R)))]$ is a subfunctor of $Hom(Spec(-),Spec(S))$ and it is of the form $Hom^{I}(Spec(A),Spec(S))=\{f\in Hom(Spec(A), Spec(S))|f^\star(I)A=A\}$ for some ideal $I\subset S$.
$F$ is a scheme functor if $F$ is sheaf in zariski topology and for all field $k$ there exists open sets $x_i\in F(R_i)$ s.t. $F(k)=\cup_i (x_i)_\star(Hom(Spec(k), Spec(R_i))$.
$\textbf{Q:}$ Why is $F(k)=\cup_i (x_i)_\star(Hom(Spec(k), Spec(R_i))$ required to check $F(R)=Hom(Spec(R), X)$ for scheme $X$?(i.e. $F\to Hom(-, X)$ is equivalence of functors.) This seems to assemble pointwise information only. Furthermore $Spec(R_i)$ is basically forming covering of $X$ and the only part of information missing here is gluing information which I think the glueing part is identified by checking at $k-$points overlapped part.