= Solution
A <morphism of algebraic varieties> is a map regular in local affine coordinates: for each point of the source, choose affine charts containing it and its image, and require each target coordinate to pull back locally to a quotient of <polynomials> with a denominator nonzero at that point. The map is continuous in the <Zariski topology> and pulls back <regular functions> on every target open <set> to <regular functions> on its inverse image.
Equivalently, one can take Zariski <continuity> and this local regular-function pullback property as the definition. In particular, a morphism between affine varieties is specified by globally regular coordinate <functions>, with values satisfying the defining equations of the target. Being an arbitrary set-theoretic <function>, or being merely Zariski continuous, is not sufficient.
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