The Universal Property of Projective Space

Question

Since the theories of affine and projective geometry are specified by certain axioms, we can consider the category $\mathcal{A}$ and $\mathcal{P}$ of affine and projective geometries, whose morphisms are maps preserving colinearity. Given an affine geometry $G$, there is a natural way to extend the geometry to a projective geometry $G_\infty$ by adding a single `line at infinity' on which all parallel lines intersect. The association $G \mapsto G_\infty$ is trivially a functor, because a map $G \to H$ induces a map $G_\infty \to H_\infty$, and these maps behave properly under composition. My question is whether there is a universal property which uniquely describes the projective extension of an affine geometry up to isomorphism, like we see for the universal properties of the free group over a set, or the Stone-Cech compactification.