I've been looking over Lurie's DAG X, and he introduces a combinatorial construction called the *twisted arrow construction* for simplicial sets that generalizes the following ordinary categorical notion:


The Yoneda embedding $\mathcal{C}\to \widehat{\mathcal{C}}$ is adjunct under the hom-tensor adjunction to the functor $$\operatorname{Hom}:\mathcal{C}^{\operatorname{op}}\times\mathcal{C}\to \operatorname{Set}$$. Composing $\operatorname{Hom}$ with the inclusion of $\iota:\operatorname{Set}\hookrightarrow\operatorname{Cat}$ gives us a functor to $\operatorname{Cat}$, and to this functor we can apply the Grothendieck construction to produce a discrete fibration $$\pi:\int^{\mathcal{C}^{\operatorname{op}}\times\mathcal{C}} \operatorname{Hom} \to \mathcal{C}^{\operatorname{op}}\times\mathcal{C}$$


We call the total space of this fibration $\operatorname{Tw}(\mathcal{C})$, the twisted arrow category.


In chapter 4 of DAG X, Lurie gives a concrete and combinatorial description of how to generalize this to any simplicial set $X$ by constructing it as the pullback $\sigma^\ast X$ where $$\sigma: \Delta \to \Delta$$ sends $$[n] \mapsto [n]^{\operatorname{op}} \boxplus [n]$$ where $\boxplus$ is the ordinal sum/join (the faces and degeneracies map to the arrows in a canonical way that is induced by the join and the $\operatorname{op}$, so I am omitting them for the sake of brevity). 


In order to understand things better in the case of a 2-category and maybe come up with a combinatorial description to extend to higher categories, I'm wondering if anyone has seen or could describe that $2$-grothendieck construction assodciated with the $\operatorname{Hom}$ functor in the case of strict $2$-categories. I'm curious if maybe something like just the (horizontal) join could possibly work to generalize it to the case of $\Theta\_2$, which plays the role in $2$-categories that $\Delta$ plays in $1$-categories.


Thanks!



Twisted-arrow construction for 2-categories
While it is more explicit and combinatorial, this probably isn't precisely what you're looking for. It does seem relevant to your question, though. A year or so ago I worked out a 2-categorical analogue of the $(\infty,1)$ twisted arrow construction (I'm working with $(\infty,1)$-categories arising as Joyal fibrant replacements of the Duskin nerves of 2-categories). I initially tried the same approach Mike Schulman suggested in the comments, i.e., using Buckley's 2-categorical Grothendieck construction, but I found that a simpler construction was sufficient for my purposes. It's a quite straightforward generalization of the commutative (up to natural isomorphism) diagram
$$
\require{AMScd}
\begin{CD}
\operatorname{Cat} @>{N}>> \operatorname{Set}\_\Delta \\
@V{Tw}VV @VV{Tw}V \\
 \operatorname{Cat} @>>{N}> \operatorname{Set}\_\Delta
\end{CD}
$$
to 2-categories. The construction is a little *ad-hoc*, but the basic idea is the following: From a 2-category 
$\mathscr{C}$, construct a new 2-category $Tw\_2(\mathscr{C})$ by letting 


* Objects of $Tw\_2(\mathscr{C})$ are morphisms of $\mathscr{C}$
* 1-morphisms in $Tw\_2(\mathscr{C})$ from $f$ to $f^\prime$ consist of diagrams
$$
 \begin{array}{c c c }
 A & \overset{f}{\rightarrow} & B \\
 h \downarrow\hspace{6pt} & \searrow & \hspace{6pt}\uparrow k \\
 A^\prime & \underset{f^\prime}{\to} & B^\prime\\
\end{array}
$$
and 
$$
 \begin{array}{c c c }
 A & \overset{f}{\rightarrow} & B \\
 h \downarrow\hspace{6pt} & \nearrow& \hspace{6pt}\uparrow k \\
 A^\prime & \underset{f^\prime}{\to} & B^\prime\\
\end{array}
$$
where each triangle commutes up to a (not nec. invertible) 2-morphism, and the 2-morphisms satisfy the obvious commutativity condition. (Note that, by definition, these are the 3-simplices of the Duskin nerve of $\mathscr{C}$.)
* 2-morphisms are given by (appropriately coherent) natural transformations of diagrams which are the identity on $f$ and $f^\prime$.


The resulting category $Tw\_2(\mathscr{C})$ is only lax unital. However, with the appropriate nerve constructions, the diagram 
$$
\require{AMScd}
\begin{CD}
\operatorname{2Cat} @>{N}>> \operatorname{Set}\_\Delta \\
@V{Tw\_2}VV @VV{Tw}V \\
 \operatorname{Lax2Cat} @>>{N}> \operatorname{Set}\_\Delta
\end{CD}
$$
commutes up to natural isomorphism. 


There is an obvious functor
$$
Tw\_2(\mathscr{C}) \to \mathscr{C}\times \mathscr{C}^{\operatorname{op}},
$$
though I haven't looked at its fibrancy properties yet.


Disclaimer: This is my first answer here, please let me know if I have ignored some protocol out of inexperience.


