If $\mathcal{C}$ is a symmetric monoidal $(\infty,1)$-category with duals, then there should be a functor $$ d: \mathcal{C} \longrightarrow \mathcal{C}^{op} $$ such that $d(x)$ is dual to $x$ for all objects $x \in \mathcal{C}$. How can one construct such a functor?

Of course, the above is only a very weak formulation of what one would actually expect. In fact, it should be possible to choose $d$ such that there is a coherent homotopy $d^{op} \circ d \simeq \operatorname{Id}_{\mathcal{C}}$. By 'coherent' I mean that $\mathcal{C}$ should have homotopy fixed-point data for the involution $$ (\ )^{op}: \mathit{Cat}_\infty^{\otimes} \longrightarrow \mathit{Cat}_\infty^{\otimes}. $$ And while I'm on it, let me give the most general version of the quesion:

Is there a canonical (or maybe even essentially unique) trivialisation of the $(\ )^{op}$ involution when restricted to the full subcategory $\mathit{Cat}_\infty^{\otimes,d}\subset \mathit{Cat}_\infty^{\otimes}$ of those symmetric monoidal $(\infty,1)$-categories in which every object is dualisable?

I am aware of a solution to this question in the $1$-categorical context; it goes as follows:

For a symmetric monoidal $1$-category $\mathcal{C}$ one constructs a category $D(\mathcal{C})$ where objects are tuples $(x,y,e,c)$ with $x,y$ two objects of $\mathcal{C}$ and $(e,c)$ is an evaluation-coevaluation pair that exhibits them as dual. The morphisms $\varphi:(x,y,e,c) \to (x',y',e',c')$ in this category are pairs $(f:x \to x',g:y' \to y)$ such that $f$ is dual to $g$ with respect to the duality data specified. This category admits canonical projections $$ \mathcal{C} \longleftarrow D(\mathcal{C}) \longrightarrow \mathcal{C}^{op}. $$ If $\mathcal{C}$ has duals, one can use essential uniqueness of duals to show that both projections are categorical equivalences. Using this construction it should be rather straight-forward to show all claims made above for the special case of $1$-categories. (Except for uniqueness of the trivialisation.)

Now the problem is that I have no idea how to generalise this proof to the $\infty$-categorical situation. For instance, for the two morphisms $f$ and $g$ to be dual would then be additional structure instead of a property.

I would be happy to use the cobordism hypothesis in dimension $1$, if that is of any help. (It can be used to construct $d$ on the maximal subgroupid $\mathcal{C}^\sim \subset \mathcal{C}$, but it seems hard to use it to say anything about duals of non-invertible morphisms. As a side-note: Where can I find a proof or even a proof sketch of the cobordism hypothesis in dimension $1$?)

**I'm interested in answers at any stage of generality.**