I am reading Strickland - Formal schemes and formal groups. For a functor $X\colon \operatorname{Rings}\to \operatorname{Sets}$, he defines (2.14) the category of $\operatorname{Points}(X)$ in the following way:

Objects: $(R,x)$ with $x \in X(R)$.
Morphisms: ring morphisms $f\colon R\to S$ s.t $X(f)(x)=y$.

In remark 2.15 he claims that the category $\mathcal{X}_X$ of affine schemes $Y$ over an affine scheme $X$ and the category of representable functors $Y'\colon \operatorname{Points}(X)\to \operatorname{Sets}$ are equivalent using $Y(S)= \coprod_{z \in X(S)} Y'(S,z)$. I can't really understand how the equivalence follows. Would appreciate any help.

P.S. I understood the equivalence of (a) and (c)—this is usually referred to as the description of a scheme using its functor of points.