[001E] Definition 4.2.1·a (Generic object).

Let $E$ be a cartesian fibration over $B$; a generic object for $E$ is defined to be an object $\bar{u}\in \TotCat{E}$ such that for any $\bar{z}\in \TotCat{E}$ there exists a cartesian map $\bar{z}\to \bar{u}$.

Warning. Our terminology differs from that of Jacobs (Categorical Logic and Type Theory, 1999); what we refer to as a generic object here is Jacobs’ weak generic object. We prefer the unqualified terminology, as generic objects in the stronger sense are very rare.