Recall [HTT, Definition 5.2.2.1]:
Definition 5.2.2.1. Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories. An adjunction between $\mathcal{C}$ and $\mathcal{D}$ is a map $q\colon\mathcal{M}\to\Delta^{1}$ which is both a Cartesian fibration and a coCartesian fibration together with equivalences $\mathcal{C}\to\mathcal{M}_{\{0\}}$ and $\mathcal{D}\to\mathcal{M}_{\{1\}}$.
This definition comes from the corresponding fact in ordinary category theory, which is phrased in the same way but with $[1]$ in place of $\Delta^{1}$ (incidentally, is there a reference containing a proof of this fact?)
Question: Is there an analogous result in $2$-category theory, relating [lax,oplax,pseudo,$2$-][bi/] adjunctions to fibrations $\mathcal{C}\to[1]$?
