Functor of points

One idea which has been coming up a lot in the SMSTC course on Algebraic Geometry, lectured by Clark Barwick, has been the idea of the functor of points. Here, we’ll try and motivate that definition with some examples. Throughout, we work over an algebraically field $k$, and so schemes and morphisms are relative to $k$. Let $X$ be a scheme. Associated to this, we have a functor $h_X : \mathrm{Sch}/k^{\mathrm{op}} \to \mathrm{Sets}$, given by $$h_X(Y) = \mathrm{Mor}(Y, X)$$ and this is called the functor of points.