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.



(Co)limits of locally cartesian closed categories
In [this preprint](https://arxiv.org/abs/1610.08459) Isaev describes a combinatorial model category which presents the $(\infty, 1)$-category of finitely complete $(\infty, 1)$-categories. The underyling category is the category of simplicial sets equipped with collections of distinguished cones over diagrams indexed by finite simplicial sets. Fibrant-cofibrant objects are precisely quasicategories with finite limits such that the marked cones are exactly the limit cones. One can also choose the notion of a marking to include witnesses for internal hom-objects in the slices (or dependent products if you prefer). This yields a combinatorial model category whose fibrant-cofibrant objects are LCC quasicategories and thus the associated $(\infty, 1)$-category is locally presentable.


Also, note that the fortegful functor to quasicategories (I wouldn't call that an inclusion) does not preserve colimits. For example the initial LCC quasicategory has one (terminal) object one while the initial quasicategory is empty.


