Warning 5.0.0.4. The terminology of Definition 5.0.0.3 is not standard. Many authors use the term *fibration* or *Grothendieck fibration* for what we refer to as a *cartesian fibration* of categories, and use the term *opfibration* or *Grothendieck opfibration* for what we refer to as a *cocartesian fibration* of categories. Our motivation is to be consistent with the terminology we will use for the analogous definitions in the $\infty $-categorical setting (see ยง5.1), where it is important to distinguish between several different notions of fibration.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$