  In this article we introduce four variance flavours of cartesian 2-fibrations
of $\infty$-bicategories with $\infty$-bicategorical fibres, in the framework
of scaled simplicial sets. Given a map $p\colon \mathcal{E}
\rightarrow\mathcal{B}$ of $\infty$-bicategories, we define $p$-(co)cartesian
arrows and inner/outer triangles by means of lifting properties against $p$.
Inner/outer (co)cartesian 2-fibrations are then defined to be maps with enough
(co)cartesian lifts for arrows and enough inner/outer lifts for triangles,
together with a compatibility property with respect to whiskerings in the outer
case. By doing so, we also recover in particular the case of
$\infty$-bicategories fibred in $\infty$-categories studied in previous work.
We also prove that equivalences of such 2-fibrations can be tested fiberwise.
As a motivating example, we show that the domain projection
$\mathrm{d}\colon\mathrm{RMap}(\Delta^1,\mathcal{C})\rightarrow \mathcal{C}$ is
a prototypical example of an outer cartesian 2-fibration, where
$\mathrm{RMap}(X,Y)$ denotes the $\infty$-bicategory of functors, lax natural
transformations and modifications. We then define inner/outer (co)cartesian
2-fibrations of categories enriched in $\infty$-categories, and we show that a
fibration $p\colon \mathcal{E} \rightarrow \mathcal{B}$ of such categories is a
(co)cartesian inner/outer 2-fibration if and only if the corresponding scaled
nerve $\mathrm{N}^{\mathrm{sc}}(p)\colon \mathrm{N}^{\mathrm{sc}}\mathcal{E}
\rightarrow \mathrm{N}^{\mathrm{sc}}\mathcal{B}$ is a fibration of this type
between $\infty$-bicategories.
