一句话：将 Hofmann–Streicher 宇宙提升推广为相对版本，并以伪伴随和 2-双纤维化重新刻画。
Hofmann 与 Streicher 在 1997 年给出了将 Grothendieck 宇宙从集合范畴提升到小范畴上集合值预层范畴的显式构造。Awodey 后来用范畴 nerve 这一函子性分析解释了该构造：对通用小离散纤维化作用，可得到宇宙 Hofmann–Streicher 提升的泛型族。受此启发，本文定义 Hofmann–Streicher 提升的相对版本，其核心是与给定纤维化后复合所得 2-函子的右伪伴随；作者进一步构造了一个新的纤维化 2-双纤维化，其中 opcartesian 与 cartesian 提升由这些伪伴随产生。
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.