I would like to formulate the cobordism hypothesis for general tangential structure as a statement about adjoint $(\infty,1)$-functors.

For a space $Y$ with an action of $O(n)$ let $X=Y\times_{O(n)} EO_n$ and let $\zeta$ be the pullback of the universal bundle under the map $X \to BO_n$. Then I will write $\mathit{Bord}_n^{\ Y}$ for what is usually called $\mathit{Bord}_n^{(X,\zeta)}$. I believe that using the notation from Lurie's paper we then have $\widetilde{X} \simeq Y$.

Therefore Theorem 2.4.18 in the same paper states that for $\mathcal{C}$ a symmetric monoidal $(\infty,n)$-category with duals $$ \operatorname{Fun}^\otimes(\mathit{Bord}_n^{\ Y}, \mathcal{C}) \simeq \operatorname{Map}_{O(n)}(Y,\operatorname{Fun}^\otimes(\mathit{Bord}_n^{fr}, \mathcal{C})). $$ Here I used that by the cobordism hypothesis for framed manifolds we have that $$ \operatorname{Fun}^\otimes(\mathit{Bord}_n^{\ \!\operatorname{fr}}, \mathcal{C}) \xrightarrow{\ \ \sim\ \ } \mathcal{C}^{\sim}. $$ And the $O(n)$ action is the one coming from $\mathit{Bord}_n^{fr}$.

My question is now if the cobordism hypothesis implies (or is equivalent to) the statement that there is an adjunction of $(\infty,1)$-functors $$ \mathit{Bord}_n^{(\underline{\quad})}: O(n)\text{-}\mathit{Spaces} \rightleftarrows \mathit{Cat}_{(\infty,n)}^\otimes : \operatorname{Fun}^\otimes(\mathit{Bord}_n^{\ \!\operatorname{fr}}, \mathcal{C}). $$

If I'm not mistaken this should follow from the above, but I am unsure as to how many technical details one has to check to actually prove this. Has this way of stating it maybe already appeared somewhere so I could cite it?

Also, the main reason why I would like to have this statement is because I would like to know that $\mathit{Bord}_n^{(\underline{\quad})}$ preserves homotopy colimits. Is there a way to see that more immediately, not invoking the (almighty) cobordism hypothesis?