= Solution
A <morphism of varieties>[morphism] $F:X\to Y$ is a <continuous> map in the Zariski topologies such that for every open $V\subseteq Y$ and every <regular function> $h$ on $V$, the <pullback> $h\circ F$ is regular on $F^{-1}(V)$.
Equivalently, near every <point>, choose <affine charts> of source and target; the target coordinates of $F$ must then be <regular functions> of the source coordinates. These local conditions agree on overlaps. In particular a <morphism of varieties>[morphism] is defined at every <point> of its source, not merely on a <dense> <open subset>.
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