In 1997, Hofmann and Streicher introduced an explicit construction to lift a Grothendieck universe from the category of sets into the category of set-valued presheaves on a small category. More recently, Awodey presented an elegant functorial analysis of this construction in terms of the categorical nerve, the right adjoint to the functor that takes a presheaf to its category of elements; in particular, the categorical nerve's functorial action on the universal small discrete fibration gives the generic family of the universe's Hofmann-Streicher lifting. Inspired by Awodey's analysis, we define a relative version of Hofmann-Streicher lifting in terms of the right pseudo-adjoint to the 2-functor given by postcomposition with a fibration. Finally, we construct a new 2-bifibration of fibrations in which the opcartesian and cartesian lifts arise from these pseudo-adjunctions.

翻译：1997年，Hofmann 和 Streicher 提出了一种显式构造，用于将 Grothendieck 宇宙从集合范畴提升到小范畴上的集合值预层范畴。最近，Awodey 利用范畴神经（即将预层映射到其元素范畴的函子的右伴随）对这一构造进行了优雅的函子性分析；特别地，范畴神经在通用小离散纤维化上的函子性作用给出了该宇宙的 Hofmann-Streicher 提升的泛族。受 Awodey 分析的启发，我们利用由纤维化后复合给出的 2-函子的右伪伴随，定义了 Hofmann-Streicher 提升的一个相对版本。最后，我们构造了一个新的纤维化的 2-双纤维化，其中协笛卡尔和笛卡尔提升均源于这些伪伴随。