Andrew Slattery, Jonathan Sterling
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.