Hofmann-Streicher lifting of fibred categories : Dedicated to the memory of Thomas Streicher (1958-2025)

In 1997, Hofmann and Streicher introduced an explicit construction to lift a Grothendieck universe ${\mathcal{U}}$ from Set into the category of Set-valued presheaves on a ${\mathcal{U}}$-small category B. 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 ${\mathcal{U}}$-small discrete fibration gives the generic family of ${\mathcal{U}}$'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 Fib<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</inf> → Fib<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</inf> given by postcomposition with a fibration $p:A \to B$.