equivalences in/of $(\infty,1)$-categories
The universal fibration of (∞,1)-categories is the generalized universal bundle of $(\infty,1)$-categories in that it is Cartesian fibration
over the opposite category of the (∞,1)-category of (∞,1)-categories such that
its fiber $p^{-1}(C)$ over $C \in (\infty,1)Cat$ is just the $(\infty,1)$-category $C$ itself;
every Cartesian fibration $p : C \to D$ arises as the pullback of the universal fibration along an (∞,1)-functor $S_p : D \to (\infty,1)Cat^{op}$.
Recall from the discussion at generalized universal bundle and at stuff, structure, property that for n-categories at least for low $n$ the corresponding universal object was the $n$-category $n Cat_*$ of pointed $n$-categories. $Z$ should at least morally be $(\infty,1)Cat_*$.
…see section 3.3.2 of HTT
The universal fibration of $(\infty,1)$-categories restricts to a Cartesian fibration $Z|_{\infty Grpd} \to \infty Grpd^{op}$ over ∞Grpd by pullback along the inclusion morphism $\infty Grpd \hookrightarrow (\infty,1)Cat$
The ∞-functor $Z|_{\infty Grpd} \to \infty Grpd^{op}$ is even a right fibration and it is the universal right fibration.
It is closely related to the object classifier in ∞Grpd.
The following are equivalent:
An ∞-functor $p : C \to D$ is a right Kan fibration.
Every functor $S_p : D \to (\infty,1)Cat$ that classifies $p$ as a Cartesian fibration factors through ∞-Grpd.
There is a functor $G_p : D \to \infty Grpd$ that classifies $p$ as a right Kan fibration.
This is proposition 3.3.2.5 in HTT.
For concretely constructing the relation between Cartesian fibrations $p : E \to C$ of (∞,1)-categories and (∞,1)-functors $F_p : C \to (\infty,1)Cat$ one may use a Quillen equivalence between suitable model categories of marked simplicial sets.
For $C$ an (∞,1)-category regarded as a quasi-category (i.e. as a simplicial set with certain properties), the two model categories in question are
the projective global model structure on simplicial presheaves on $[C,SSet]$ – this models the (∞,1)-category of (∞,1)-functors $(\infty,1)Func(C,(\infty,1)Cat)$.
the covariant model structure on the over category $SSet/C$ – this models the $(\infty,1)$-category of Cartesian fibrations over $C$.
The Quillen equivalence between these is established by the relative nerve? construction
By the adjoint functor theorem this functor has a left adjoint
For $p : E \to C$ a left Kan fibration the functor $F_p(C) : C \to SSet$ sends $c \in Obj(C)$ to the fiber $p^{-1}(c) := E \times_C \{c\}$
(See remark 3.2.5.5 of HTT).
The universal fibration as such is discussed in section 3.3.2 of
The concrete description in terms of model theory on marked simplicial sets is in section 3.2. A simpler version of this is in section 2.2.1