Definition 6.16.
Let \(C\) be a category equipped with a Grothendieck topology \(\tau\). Let \(f\colon X \to Y\) be a morphism in \(\Shv_{\tau}(C)\) and assume that \(Y = y_{\tau}(Y')\) lies in the image of the sheafified Yoneda functor. We say that \(f\) admits local sections if there exists a covering family \(\{U_i \to Y'\}_{i \in I}\) of \(Y'\) such that the base change
\[X \times_{y_{\tau}(Y')} y_{\tau}(U_i) \to y_{\tau}(U_i)\]
admits a section for every \(i \in I\).
If \(f\colon X \to Y\) is an arbitrary morphism in \(\Shv_{\tau}(C)\), we say that \(f\) admits local sections if its base change along every map \(y_{\tau}(Y') \to Y\) from a sheafified representable admits local sections.