Translation between the functor-of-point and locally ringed space approach to schemes

Question

Functorially, we have the following defintions:

1. An affine scheme $U$ is a representable (covariant) functor $\mathsf{CRing} \to \mathsf{Sets}$.
2. A scheme $X$ can be defined as a locally representable cosheaf of sets on $\mathsf{CRing}$.

How then, can I recover the "traditional" definition that $U = (\text{Spec} A, \mathcal{O}_{\text{Spec} A})$ and $X = (|X|, \mathcal{O}_X)$ so I could look at the rings of functions above $U$ and $X$, preferably functorially ?