Hofmann-Streicher lifting of fibred categories

In 1997, Hofmann and Streicher introduced an explicit construction to lift a Grothendieck universe $\mathcal{U}$ from $\mathbf{Set}$ into the category of $\mathbf{Set}$-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 $\mathbf{Fib}_{A}\to\mathbf{Fib}_{B}$ given by postcomposition with a fibration $p\colon A\to B$.