Given a symmetric monoidal $(\infty,n)$-category $\mathcal{D}$, one obtains a symmetric monoidal $(\infty,n-1)$-category $\Omega \mathcal{D}$ by taking $\Omega \mathcal{D}= End_{\mathcal{D}}(1_{\mathcal{D}})$, where $1_{\mathcal{D}}$ is the unit object of $D$. Conversely, given a symmetric monoidal $(\infty,n-1)$-category $\mathcal{C}$, one obtains a symmetric monoidal $(\infty,n)$-category $B \mathcal{C}$ by defining $B \mathcal{C}$ to be the $(\infty,n)$-category with a single object with $\mathcal{C}$ as category of morphisms.

How are $\Omega$ and $B$ related? Clearly $\Omega B\mathcal{C}$ is again $C$ and $B\Omega\mathcal{D}$ is the full monoidal subcategory of $\mathcal{D}$ generated by the object $1_{\mathcal{D}}$. But maybe there is more: in a recent discussion on this subject with Alessandro Valentino we came to conjecturing that, mimicing what happens between topological monoids and (pointed) topological spaces (see, e.g. http://arxiv.org/abs/1203.4978), $\Omega$ and $B$ should be adjoint in a suitable sense. More precisely we expect one should have

$$ Fun^{\otimes}(B\mathcal{C},\mathcal{D}) \cong Fun^{\otimes}(\mathcal{C},\Omega\mathcal{D}) $$

where $Fun^{\otimes}$ stands for ``symmetric monoidal functors''. But we have so far been unable to prove (or disprove) such a statement nor to locate it in the literature.

Any suggestion in either direction will be appreciated :)