Let $\underline{\mathbb{A}}^r$ be the functor from $\textbf{Schemes}$ to $\textbf{Sets}$ which associates to each scheme $S$ the set of morphisms $\bigoplus_{k=1}^rO_S \to O_S$ ($O_S$ is the structure sheaf of $S$). Let $\mathbb{A}^r$ be the scheme $Spec(\mathbb{Z}[X_1, \dots, X_r])$.
Assume it is given that whenever $X$ is a scheme the restriction of $\underline{\mathbb{A}}^r$ to $Top(X)$ (which I presume is the subcategory of $\textbf{Schemes}$ consisting of the open subschemes of $X$ and the open immersions between them) is a sheaf of sets on $X$.
Assume further that the restriction of the representable functor $h_{\mathbb{A}^r} := Mor(\_, \mathbb{A}^r)$ to Top(X) is also a sheaf of sets on $X$.
If we wish to show that $\underline{\mathbb{A}}^r$ is representable by $\mathbb{A}^r$, why does it suffice to show that there is an isomorphism between $\mathbb{A}^r$ restricted to the category $\textbf{Aff}$ of affine schemes and $\underline{\mathbb{A}}^r$ restricted also to $\textbf{Aff}$? This is just stated at some point in a proof in my Schemes course.
I vaguely see that perhaps the fact that every scheme is a bunch of affine schemes glued together, and that 'sheaves allow us to glue', might help - but given that $Top(X)$ does not appear to be a full subcategory of $\textbf{Schemes}$ in general I do not quite follow the logic. It is stated that this fact follows from the two facts stated above about restrictions to $Top(X)$.
What am I misunderstanding, here?