I am trying to prove Exercise 1.5.4 from Brian Conrad's course notes on abelian varieties. Let $S = \mathrm{Spec} R$, let $X, Y$ be $S$-schemes, and let $h_{\mathrm{aff}, X}(-)$ denote the Hom functor $\mathrm{Hom}_{S}(-, X)$ on the category of affine $S$-schemes (hence not representable if $X$ is not affine). The goal of the exercise is to establish a bijection $\mathrm{Hom}_{S}(X, Y) \cong \mathrm{Nat}(h_{\mathrm{aff}, X},h_{\mathrm{aff}, Y})$, strengthening the usual Yoneda lemma when the base scheme is affine. This is useful because it is often easier to consider functors of points from affine schemes rather than arbitrary schemes.
The usual proof of Yoneda does not seem to work since the functor $h_{\mathrm{aff}, Y}(-)$ does not explicitly include the data of morphisms $\mathrm{Hom}_{S}(X, Y)$ if $X$ is not affine. I think I have succeeded in proving the exercise in the case where $X$ is separated over $S$. If $\eta$ is a natural transformation, we may cover $X$ by open affines $U_i$ and consider the induced maps $\eta_{U_i}: \mathrm{Hom}_{S}(U_i, X) \to \mathrm{Hom}_{S}(U_i, Y)$. The open embeddings $U_i, U_j \hookrightarrow X$ map to the same element of $\mathrm{Hom}_{S}(U_i \cap U_j, X)$, hence by naturality of $\eta$ the induced morphisms $U_i \to Y, U_j \to Y$ agree on $U_i \cap U_j$, hence glue to a unique morphism $X \to Y$. However, we can only make this argument if $U_i \cap U_j$ is known to be affine; one way to guarantee this if $X$ is separated over $S$ or some other affine scheme.
Is the separatedness hypothesis necessary, or is there a way to get around it?
