= Affine test schemes determine scheme morphisms
{title2=$\operatorname{Hom}(X,Y)\cong\operatorname{Nat}(g_X,g_Y)$}
Restricting the <functor of points> to <affine schemes> gives a <fully faithful> <functor> from <schemes> to covariant set-valued <functors> on <commutative rings>. Given a <natural transformation>, apply it to the inclusions of an <affine open cover> of $X$. <Naturality> on <affine open subschemes> of overlaps makes the resulting maps to $Y$ agree, so they glue. For an arbitrary map from an <affine scheme> to $X$, cover its inverse images of these charts by <affine open subschemes> and apply <naturality> again. This proves that the glued map induces the prescribed <natural transformation>, and uniqueness follows from the original cover. In particular, the restricted <functor of points> determines the <scheme> up to an <isomorphism of schemes>.
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