For every scheme $X$, we can attach the functor $Hom(-,X): Sch \to Set $ and then Yoneda Lemma tells us $Hom(-,X)\cong Hom(-,Y)$ if and only if $X\cong Y$.
Now suppose $\mathcal A$ denote the full subcategory of $Sch$ consisting of integral, Noetherian schemes of dimension $1$. For a fixed algebraically closed field $k$, let $\mathcal B_k$ denote the full subcategory of $Sch_k$ (so the morphisms are morphism of Schemes commuting with the structure morphism of $k$) consisting of integral $k$-schemes of finite type of dimension $1$ and let $\mathcal C_k$ denote the full subcategory of $Sch_k$ consisting of separated, integral $k$-schemes of finite type of pure dimension $1$.
I have three related questions:
(1) If $X,Y$ are integral, Noetherian schemes of finite dimension such that as functors $\mathcal A \to Set$, we have an isomorphism of functors $Hom(-,X)\cong Hom(-,Y)$, then is it true that $X \cong Y$ ?
(2) If $X,Y$ are integral $k$-schemes of finite type such that as functors $\mathcal B_k \to Set$, we have an isomorphism of functors $Hom(-,X)\cong Hom(-,Y)$, then is it true that $X \cong Y$ ?
(3) If $X,Y$ are integral, separated $k$-schemes of finite type such that as functors $\mathcal C_k \to Set$, we have an isomorphism of functors $Hom(-,X)\cong Hom(-,Y)$, then is it true that $X \cong Y$ ? If this is not true in general, then what if we also assume $X,Y$ are both quasi-projective, or both affine, or both projective ?