TITLE: (Co)limits of locally cartesian closed categories
QUESTION [6 upvotes]: Do locally cartesian closed $\infty$-categories form a presentable $\infty$-category? It seems like they should, and that the inclusion $\text{LCC}\rightarrow\text{Cat}$ preserves limits and maybe even colimits.
Here is a possible strategy for proof. There is a functor $\text{Cat}\rightarrow\text{Fun}(\Delta^1,\text{Cat})_\text{cart}$ taking an $\infty$-category $\mathcal{C}$ to the cartesian fibration $\text{Fun}(\Delta^1,\mathcal{C})\rightarrow\mathcal{C}$; this fibration classifies the functor $\mathcal{C}\rightarrow\text{Cat}$ which sends $X$ to $\mathcal{C}_{/X}$. The condition that $\mathcal{C}$ is locally cartesian closed is precisely that for all $X\rightarrow Y$, $\mathcal{C}_{/X}\rightarrow\mathcal{C}_{/Y}$ has a chain of two right adjoints (its right adjoint has a right adjoint). This should correspond to some condition on the cartesian fibration, so that $\text{LCC}$ is a pullback of $\text{Cat}\rightarrow\text{Cat}^\text{cart}$ along some subcategory of $\text{Cat}^\text{cart}$. A pullback of presentable categories should be presentable.
But I'm not sure how to handle the "chain of two right adjoints", and I wonder if I am missing an easier argument.

REPLY [5 votes]: First, a clarification: what are the morphisms of locally cartesian closed categories meant to be?
Now, a word on the "two adjoints" thing.
For every category $C$, there is a cocartesian fibration $\mathrm{cod}: C^{\Delta[1]} \to C$ with fiber $C/c$ over $c \in C$; reindexing is given by composition. $C$ has pullbacks if and only if $\mathrm{cod}$ is simultaneously a cartesian fibration (i.e. it is a bifibration); reindexing is given by pullback.
If $C$ has pullbacks, then the cartesian fibration $\mathrm{cod}$ classifies a functor $C^\mathrm{op} \to \mathsf{Cat}$. This functor is also classified by a cocartesian fibration $\overline{\mathrm{cod}}: E \to C^\mathrm{op}$ (different from $\mathrm{cod}$). In the "total category" $E$, an object over $c \in C^\mathrm{op}$ is a map $f: d \to c$ in $C$, while a morphism from $f$ to $f'$ over $\gamma: c \leftarrow c'$ is a map $c \times_d c' \to d'$. Now $C$ is locally cartesian closed if and only if the cocartesian fibration $\overline{\mathrm{cod}}$ is simultatneously a cartesian fibration. Pavlovic calls such a functor a "trifibration"; his paper is referenced in the nlab page on bifibrations linked to above.
The process of straightening a cartesian fibration and then unstraightening it to get a cocartesian fibration may sound unwieldy in the $\infty$-categorical setting, but it has been studied by Barwick, Glasman, and Nardin, and I think they give a more direct construction.
Upshot:
This allows one to check that (small) locally cartesian closed categories form a presentable category, since they lie in a pullback diagram of accessible right adjoints between such:
$\require{AMScd}
\begin{CD}
Cat^{lcc} @>>> Cat^{cart} \\
@VVV @VVV\\
Cat^{pb} @>>> Cat^{\Delta[1]}
\end{CD}$