The category of schemes sits between $Aff=CRing^\text{op}$ and its free cocompletion $\hom(Aff^\text{op},Sets)$, as the locally representable functors. Dualizing, we have $CRing\subseteq Scm^\text{op}\subseteq \hom(CRing^\text{op},Set^\text{op}).$ So is there a description of $Scm^\text{op}$ that sits on the algebraic side of the duality between algebra and geometry?
this interesting response by Zhen Lin Low about why covariant Yoneda is more natural than contravariant suggests perhaps we may not expect such a description ($Sets^\text{op}$ is not as familiar?)
