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 $X to Y$.