Allow me to waffle about a bit of topos theory, leading up to a few questions about KZ comonads and about the comprehension schema in hyperdoctrines.
Let $Pr^L$ denote the $\infty$-category of presentable $\infty$-categories, and let $Pr^L_\kappa$ denote the locally-full subcategory of $\kappa$-compactly-generated $\infty$-categories (which is also an object of $Pr^L$ -- in fact we have $Pr^L_\kappa \in Pr^L_\kappa$, which is surprising at first -- here $\kappa$ is an uncountable regular cardinal).
Observation 1: Let $\mathcal C \in Pr^L_\kappa$. Then there is an adjunction
$$ ev_1 : Pr^L_\kappa / \mathcal C {~~}^\longrightarrow_\longleftarrow ~~\mathcal C : \mathcal C / (-)$$
Moreover, the right adjoint $\mathcal C / (-)$ is fully faithful.
Here, $ev_1(F : \mathcal X \to \mathcal C):= F(1)$ where $1$ is the terminal object of $\mathcal X$; $\mathcal C / (-)$ carries $c \in \mathcal C$ to the domain functor $dom : \mathcal C / c \to \mathcal C$.
Observation 2: Let $\mathcal C \in Pr^L_\kappa$. Then the following are equivalent:
The functor $ev_1 : Pr^L_\kappa / \mathcal C \to \mathcal C$ admits a right adjoint in $Pr^L$;
The functor $\mathcal C / (-) : \mathcal C \to Pr^L_\kappa / \mathcal C$ admits a right adjoint in $CAT$;
$\mathcal C$ is an $\infty$-topos.
Proof Sketch: Asking for the covariant slice functor $\mathcal C / (-)$ to preserve colimits is asking for the contravariant slice functor to preserve limits, as a functor $\mathcal C^{op} \to CAT$. This is the usual statement of descent.
I really want Observation 2.1 to say that $Pr^L_\kappa / (-) : Pr^L \to Pr^L$ is a KZ-comonad (with counit $ev_1$) whose coalgebras in $Pr^L_\kappa$ are the $\kappa$-compactly-generated $\infty$-topoi. But this can't be right, because $Pr^L_\kappa / \mathcal C$ is not an $\infty$-topos! So even if $Pr^L_\kappa / (-)$ is a comonad, its cofree coalgebras are not $\infty$-topoi, so in general its coalgebras are not $\infty$-topoi.
Question 1: If $Pr^L_\kappa / (-)$ is not a KZ-comonad with coalgebras the $\infty$-topoi, then what sort of structure do we have here whose coalgebras are the $\infty$-topoi?
Question 2: Is there some modification of the functor $Pr^L_\kappa / (-)$ which is a KZ comonad with coalgebras the $\infty$-topoi?
Question 3: Observation 1 says that $Pr^L_\kappa / (-)$ is a hyperdoctrine on the 2-category $Pr^L$, and Observation 2.1 says that the objects satisfying Lawvere's Comprehension schema (see the last few pages here) are the $\infty$-topoi. What should I make of this?
Question 4: The extra adjoint in Observation 2.2 is some sort of generalized universe construction which I don't understand too well. There's also a version whose domain is $Pr^L_\kappa$ rather than $Pr^L_\kappa / \mathcal C$; e.g. this functor carries $Spaces$ to the classifier for $\kappa$-small objects. How can one get a handle on this functor?
