This is the one categorical level higher version of the question Delooping monoidal $\infty$-groupoids into $\infty$-categories.
The classical, bicategorical, setting.
Given a monoidal category $(\mathcal{C},\otimes,\mathbf{1}_{\mathcal{C}})$, we have a bicategory $\mathbf{B}\mathcal{C}$, categorifying the classical delooping of a monoid into a one-object category.
It is characterised by the following property: for any other bicategory $\mathcal{D}$, we have an equivalence of categories $$ \left\{ \begin{gathered} \text{pseudofunctors}\\ \mathbf{B}\mathcal{C}\to\mathcal{D} \end{gathered} \right\} \cong \left\{ \begin{aligned} &\text{pairs $(X,\phi)$ with}\\ &\,\,\,\,\,\,\,\text{- $X$ an object of $\mathcal{D}$;}\\ &\,\,\,\,\,\,\,\text{- $\phi$ a strong monoidal functor}\\ &\text{from $\left(\mathcal{C},\otimes,\mathbf{1}\right)$ to $\left(\mathsf{Hom}_{\mathcal{D}}(X,X),\right.$}\\ &\text{$\left.\circ,\mathrm{id}_{X}\right)$.} \end{aligned} \right\}. $$ For example:
- A pseudofunctor $\mathbf{B}\mathbb{F}\to\mathcal{C}$ is the same as an endomorphism $A\to A$ of $\mathcal{C}$;
- A pseudofunctor $\mathbf{B}\tau_{\leq1}\mathbb{S}\to\mathcal{C}$ is the same as an automorphism $A\to A$ of $\mathcal{C}$.
The $(\infty,2)$-categorical setting.
Preliminary Question. Given an $(\infty,2)$-category $\mathcal{C}$, is there a natural monoidal $\infty$-category structure on $\mathrm{Hom}_{\mathcal{C}}(X,X)$?
Main Question. Is there an analogous delooping construction from monoidal $\infty$-categories to $(\infty,2)$-categories, such that, similarly to the above:
- A functor from $\mathbf{B}\mathcal{C}$ to another $(\infty,2)$-category $\mathcal{D}$;
is the same as
- An object $X$ of $\mathcal{D}$ together with a functor of monoidal (?) $(\infty,1)$-categories $\mathcal{C}\to\mathrm{Hom}_{\mathcal{D}}(X,X)$,
and such that this bijection can be made into a full-fledged isomorphism of (appropriate) $\infty$-categories?
