Confusion when defining a sheaf on a basis of open sets

Question

This question relates to Perrin's Algebraic Geometry: An Introduction, specifically Chapter III, Lemma 2.1. In his definition of $\bar{\mathcal{F}}$, if $U_i$ is a basic open set which isn't contained in $U$, how can we say that $s|_{U_i} \in \mathcal{F}(U_i)$? Specifically, how are we expected to define $s|_{U_i}$ at points of $U_i \backslash U$?