Twisted-arrow construction for 2-categories 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!
 A: 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.
A: Since you started with the quasi-categorical version of the twisted arrow category (and not the complete Segal space version), maybe $\Theta_2$ isn't quite the next step. Instead, let's see what happens if we keep things as close to simplicial sets as we can- say, the complicial route a la Verity. For $n=2$ it's possible to make due with less, and we have Lurie's model of $(\infty,2)$-categories via special types of scaled simplicial sets, i.e. a simplicial set $S$ together with a collection $T \subset S_2$ of thin 2-simplices (which we think of as being 'the invertible 2-morphisms') that contain all the degenerate 2-simplices. 
If $\mathcal{C}$ is an $(\infty,2)$-category, we expect the `hom'-functor to be some type of thing like $\mathcal{C}^{op} \times \mathcal{C} \to \mathsf{Cat}_{\infty}$ where the target is the $(\infty,2)$-category of $\infty$-categories. I claim that the appropriate candidate for the twisted arrow category is... the twisted arrow category (again)!
Let me say what I mean more precisely. Given a scaled simplicial set $(S, T)$ form the marked simplicial set $(\mathsf{TwArr}(S), M)$ where $\mathsf{TwArr}(S)_n = S([n]^{op} \star [n])$ and a 1-simplex is marked if and only if the two 2-simplices making up the face of the square are thin. This thing has a projection map to $S^{op} \times S$. I claim that when $S$ is fibrant (i.e. an $(\infty,2)$-category) this is an appropriate sort of fibration which is classified by the lax functor $\mathrm{Map}(-,-): S^{op} \times S \to \mathsf{Cat}_{\infty}$ mentioned above. I'll try to sketch how that goes.
But first, as a sanity check, notice that the fiber above vertices $(X,Y)$ has 0-simplices given by 1-morphisms $X \to Y$ and $1$-simplices given by a little square. Collapsing the edges $(X=X)$ and $(Y=Y)$ leaves you with a little disk with a line through the middle- all arrows oriented left to right. Keeping track of orientations of the 2-simplices tells you that the two 2-arrows glue together to gives a 2-arrow from the bottom to the top (or top to bottom depending on how you drew this.) This 2-morphism need not be thin, so we are seeing the 'category of 2-morphisms'. Not rigorous, but a little reassuring. 


*

*Lurie proves a version of straightening/unstraightening which produces, for a scaled simplicial set $X$, a Quillen equivalence between a certain model structure on $\left(\mathrm{Set}^{+}_{\Delta}\right)_{/X}$ and a certain model structure on $\mathrm{Fun}(\mathscr{C}^{sc}(X), \mathrm{Set}_{\Delta}^{+})$ (where $\mathscr{C}^{sc}$ is a scaled version of $\mathfrak{C}$). Passing to a scaled version of the nerve, if $X$ is fibrant, produces a map of $(\infty,2)$-categories $X \to \mathsf{Cat}_{\infty}$. 

*The first condition for fibrancy here is that we have an inner fibration on the underlying simplicial sets. I'll skip saying anything about this... you could draw some pictures for $\Lambda^2_1 \subset \Delta^2$ and convince yourself that if the definition of $(\infty,2)$-category says you can 'compose' 2-morphisms in an essentially unique way then you'll be ok. And it does!

*The next condition for fibrancy is that the marked edges coincide with the locally $p$-cocartesian edges. Intuitively, suppose we have a morphism $(X,Y) \to (X', Y')$ and we've chosen a lift $X \to Y$ of the object $(X,Y)$. Then a morphism covering the given one is the data of a 1-morphism and a 2-morphism from that to the composite $X' \to X \to Y \to Y'$. The universal choice of such is to take "the actual" composite and have the 2-morphism be the "identity". Of course, we're in a weak world, so to do this for real I'd have to tell you more about the definition of an $(\infty,2)$-category.

*Finally, we need to know that $p$ restricts to a cocartesian fibration along any thin 2-simplex. To check it, you need to know that composing locally cocartesian lifts of each leg gives you a locally cocartesian lift of the 'composite'. Using the characterization of locally cocartesian lifts from before, and the assumption that you're covering a thin 2-simplex, this becomes the statement that 'thin 2-simplices compose to thin 2-simplices' i.e. that 2-equivalences are closed under composition. (Again, really you check some filling condition and so on...). 

