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Warning The terminology of Definition 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.