If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\mathbf{H}/X$. Then the category $T \mathbf{H}$ is an $\infty$-topos (see [this page](https://ncatlab.org/nlab/show/tangent+%28infinity%2C1%29-category#TangentTopos) for a proof). The question is: can this fact be generalized to other Cartesian fibrations constructed in a similar way? For example, if we have an essentially algebraic $\infty$-theory $S$, then we can define in a similar way a Cartesian fibration $p\_S : T\_S \mathbf{H} \to \mathbf{H}$ such that the fiber of $p\_S$ over $X$ is the $\infty$-category of models of $S$ in $\mathbf{H}/X$. Is it true that $T\_S \mathbf{H}$ is an $\infty$-topos?



Generalizations of tangent $\infty$-topos
This is rarely true. For example, the axioms for ∞-topoi imply that, if $T\_S\mathbf H$ is an ∞-topos, then the fibers of $p\_S$ must have [van Kampen pushouts](https://ncatlab.org/nlab/show/van+Kampen+colimit) (more generally van Kampen weakly contractible colimits). In particular, the fibers cannot be $n$-categories for any finite $n$, which rules out essentially algebraic ∞-theories that are $n$-categories. This also fails for most Lawvere theories: $A\_\infty$-spaces, simplicial commutative rings, ... For a concrete example, $\mathbb Z[x,y]=\mathbb Z[x]\otimes\_{\mathbb Z}\mathbb Z[y]$ is not a van Kampen pushout of simplicial commutative rings, since both $\mathbb Z$ and $\mathbb Z[xy]$ have the same intersections with $\mathbb Z[x]$ and $\mathbb Z[y]$.


Sufficient and almost necessary conditions for a Cartesian fibration over an ∞-topos to be an ∞-topos are described in my paper <http://www-bcf.usc.edu/~hoyois/papers/loci.pdf>. In particular, if you consider the Cartesian fibration $\mathbf E\to\mathbf H$ whose fiber over $X\in\mathbf H$ is $\mathbf H/X \otimes C$ for some fixed presentable ∞-category $C$ (which I think includes your example), then $\mathbf E$ is an ∞-topos iff weakly contractible colimits in $C$ are van Kampen. Joyal calls such an ∞-category $C$ an ∞-locus (see [his IHES talk](https://www.youtube.com/watch?v=qmCh9KwrQq8)). Presentable stable ∞-categories are examples of such.


