Extended TFT with coefficients in spans in any $\infty$-topos In the TFT classification article by Jacob Lurie (arXiv:0905.0465) the (∞,n)-category of correspondences  (there: $Fam_n$) plays a key role, whose $k$-morphisms are $k$-fold spans of $\infty$-groupoids sliced over a suitable coefficient object $\mathcal{C}$. 
Motivated by some notes we had made on "Local prequantum field theory", recently Rune Haugseng  considered in detail (arXiv:1409.0837) the evident generalization of this where the collection of $\infty$-groupoids is replaced by any other $\infty$-topos. The key properties of the construction are preserved under this generalization,  in particular this is still an (∞,n)-category with duals.
It seems plausible that, similarly, essentially all the relevant statements regarding topological field theories with coefficients in such correspondences in any $\infty$-topos will still hold. I am specifically interested in a sanity check of this for the following statement:
First of all, combining prop. 3.2.8 in arXiv:0905.0465 with theorem 2.4.18 there and applying it to the special case that $\mathcal{C}$ is an $\infty$-groupoid (with duals), yields the neat statement that unoriented "local prequantum field theories with phases in $\mathcal{C}$"
$$
  Z_L \;\colon\; Bord_n^\sqcup \longrightarrow Fam_n(\mathcal{C})^\otimes
$$
are equivalent to the choice of an $\infty$-groupoid $F$ equipped with $O(n)$-$\infty$-action and with an $O(n)$-equivariant map
$$
  L \;\colon\; F \longrightarrow \mathcal{C}
  \,,
$$
where $\mathcal{C}$ is equipped with the canonical $O(n)$-action induced via theorem 2.4.6  in the above text. An equivalent way to say this is that $L$ induces a horizontal map fitting into
$$
  \array{
      F/\!/O(n) && \stackrel{L/\!/O(n)}{\longrightarrow} && \mathcal{C}/\!/O(n)
      \\
      & \searrow && \swarrow_{canonical}
      \\
      && B O(n)
  }
$$
(where the double slash denotes homotopy quotients).
This makes it very manifest that this statement has an immediate analogue with all $\infty$-groupoids replaced by objects of any $\infty$-topos $\mathbf{H}$ (with $B O(n)$ regarded as the inverse image under the terminal geometric morphism of that $\infty$-topos  of the homotopy type of the usual classifying space). Now one considers $Fam_n^{\mathbf{H}}(\mathcal{C})$ being the $(\infty,n)$-category of $n$-fold $\mathcal{C}$-phased correspondences, now all inside $\mathbf{H}$, and so forth. And $O(n)$-$\infty$-actions on $F \in \mathbf{H}$ are equivalent to homotopy fiber sequences in $\mathbf{H}$ of the form
$F \to F/\!/O(n) \to B O(n)$. But let me maybe restrict attention to the case that $\mathcal{C}$ is still a bare $\infty$-groupoid (similarly embedded into $\mathbf{H}$ under the terminal inverse image).
Is it then still true for general $\mathbf{H}$ that monoidal $(\infty,n)$-functors
$$
  Bord_n^\sqcup \longrightarrow Fam_n^{\mathbf{H}}(\mathcal{C})^\otimes
$$
are equivalent to diagrams of the form
$$
  \array{
      F/\!/O(n) && \stackrel{L/\!/O(n)}{\longrightarrow} && \mathcal{C}/\!/O(n)
      \\
      & \searrow && \swarrow_{{canonical}}
      \\
      && B O(n)
  }
$$
in $\mathbf{H}$? 
(In case it matters, I am happy to assume that the terminal inverse image of $\mathbf{H}$ is fully faithful.)
 A: Here an argument using the assumption that $\mathbf{H}$ has an $\infty$-site $\mathcal{S}$ of definition all whose objects are étale contractible. 
The proof of prop. 3.2.8  arXiv:0905.0465 shows that for $\mathbf{H} = \infty Grpd$ the equivalence in question is natural in the choice of the unoriented bulk field theory $F/\!/O(n)$.
Let then
$$
  \mathbf{H} \stackrel{\longleftarrow}{\hookrightarrow} Func(\mathcal{S}^{op},\infty Grpd)
$$
be the reflection exhibiting $\mathcal{S}$ as an $\infty$-site of definition. 
Given $F \in \mathbf{H}$ equipped with $O(n)$-$\infty$-action (throughout $O(n)$ denotes the homotopy type of the topological group $O(n)$, regarded as a group object in constant $\infty$-stacks)
write for each $U \in \mathcal{S}$
$$
   F_U := \mathbf{H}(U,F) \in \infty Grpd
$$
for its value on $U$. Observe that setting
$$
   F_U/\!/O(n) := \mathbf{H}(U,F/\!/O(n)) \in \infty Grpd
$$
exhibits an $O(n)$-$\infty$-action on $F_U$ since $\mathbf{H}(U,-)$
preserves $\infty$-limits and since by assumption on the site $\mathcal{S}$ we have
$\mathbf{H}(U,B O(n)) \simeq B O(n)$, so that the homotopy fiber sequence
$$
   F \to F/\!/O(n) \to B O(n)
$$
which exhibits the $\infty$-action of $O(n)$ on $F$ (here I am using arXiv:1207.0248) naturally induces a system of homotopy fiber sequences
$$
  F_U \to F_U/\!/O(n) \to B O(n)
$$
exhibiting $O(n)$-$\infty$-actions on each $F_U$.
It follows that a diagram
$$
  \array{
    F/\!/O(n) && \stackrel{L/\!/O(n)}{\longrightarrow} && \mathcal{C}/\!/O(n)
    \\
    & \searrow && \swarrow
    \\
    && B O(n)
  }
$$
in $\mathbf{H}$ is equivalent to an $\infty$-sheaf of diagrams of the form
$$
  U \;\;\;
  \mapsto
  \;\;\;
  \left\{
  \array{
    F_U/\!/O(n) && \longrightarrow && \mathcal{C}/\!/O(n)
    \\
    & \searrow && \swarrow
    \\
    && B O(n)
  }
  \right\}
$$
in $\infty \mathrm{Grpd}$. Now the statement of the proposition for $\mathbf{H} = \infty \mathrm{Grpd}$ applies objectwise for each $U$, and since it is natural in $F_U$ (with its action) it is also natural in $U$, and 
so the above is equivalent to the $\infty$-sheaf ($(\infty,n)$-sheaf)
of local unoriented-topological field theories in $\infty \mathrm{Grpd}$:
$$
  U \;\;\;
  \mapsto
  \;\;\;
  \left\{
  \array{
    && (Fam_n(\mathcal{C}))^{\otimes}
    \\
    & {}^{}\nearrow & \downarrow
    \\
    Bord_n^\sqcup &\underset{F_U/\!/O(n)}{\longrightarrow}& (Fam_n)^\otimes
  }
  \right\}
  \,.
$$
But this is equivalently a field theory
$$
  \array{
    && (Fam_n^{\mathbf{H}}(\mathcal{C}))^{\otimes}
    \\
    & {}^{}\nearrow & \downarrow
    \\
    Bord_n^\sqcup &\underset{F/\!/O(n)}{\longrightarrow}& (Fam_n^{\mathbf{H}})^\otimes
  }
$$
with coefficients in $\mathbf{H}$.
