Let $X = (X, \tau)$ be a topological space and $\mathcal{F}: \tau \to \mathbf{Rings}$ be a presheaf of rings on $X$. Since the rings $\mathcal{F}(U)$, with $U \in \tau$, are often thought as rings of functions, say $\mathcal{F}(U) = \{\phi \mid \phi: U \to R\}$, with $R \in \mathbf{Rings}$ fixed, I would like to define the domain of a section $\phi$ of $\mathcal{F}$ by setting $\mathrm{dom}\,\phi = U$, where $U$ is such that $\phi \in \mathcal{F}(U)$. But this definition makes sense only under the assumption that such a $U$ is unique. And it seems to me that this assumption is not backed up by the definition of a presheaf. So, may a section $\phi$ of a presheaf $\mathcal{F}: \tau \to \mathbf{Rings}$ belongs to $\mathcal{F}(U) \cap \mathcal{F}(V)$, for some $U,V \in \tau$ with $U \neq V$? What about a section of a sheaf?
PS: I am using the notes http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2014/main.pdf as main reference.