# (when) is a map of schemes determines by values on Zariski dense subset?

The motivation for this question is to justify/understand a particular abuse of notation in the world of group schemes, namely people write things like $$m:(g_1,g_2) mapsto g_1g_2$$ when discussing the multiplication or action maps $$G times_k G to G$$ when $$G$$ is a group scheme over $$k$$. This bothers me because over a general base field $$k$$, identifying points in the fibered product $$G times_k G$$ as pairs $$(g_1,g_2)$$ only makes sense on $$k$$-rational points.

However, suppose we are in the situation where the $$G(k) hookrightarrow G$$ is Zariski dense, and $$G$$ is a variety, so in particular (geometrically) reduced, separated, etc. Then I would think that a map on $$k$$-rational points, given by polynomials, determines uniquely a map of schemes.

So the question is, suppose $$X$$ is a variety of a general base field $$k$$ such that $$X(k)$$ is Zariski dense in $$X$$. Then does a regular map $$X(k) to Y$$, $$Y$$ separated, determine uniquely a morphism $$X to Y$$.