Inverting a suspension object in a stable monoidal category

Suppose we are given a cocomplete closed symmetric monoidal stable $$(\infty,1)$$-category $$\mathcal{C}$$ with suspension $$\Sigma$$, and let $$X \in \mathcal{C}$$ be dualizable. I'd like to create a new stable cocomplete symmetric monoidal category $$\mathcal{C}[\Sigma X^{-1}]$$ together with a symmetric monoidal exact and continuous functor $$\mathcal{C} \to \mathcal{C}[\Sigma X^{-1}]$$ with the following property: For any symmetric monoidal stable category $$\mathcal{D}$$,the induced functor $$Fun_{ex,cont}^{\otimes}(\mathcal{C}[\Sigma X^{-1}], \mathcal{D}) \to Fun_{ex,cont}^{\otimes}(\mathcal{C}, \mathcal{D})$$ induces an equivalence of $$Fun_{ex,cont}^{\otimes}(\mathcal{C}[\Sigma X^{-1}], \mathcal{D})$$ with the full subcategory of $$Fun_{ex,cont}^{\otimes}(\mathcal{C}, \mathcal{D})$$ consisting of functors $$\mathcal{C} \to \mathcal{D}$$ that send $$\Sigma X$$ to an invertible object in $$\mathcal{D}$$. Here the notation $$Fun_{ex,cont}^{\otimes}(\mathcal{C},\mathcal{D})$$ denotes the category of monoidal functors that are both exact and continuous.

My idea was to localize at the set of maps $$(\Sigma (\iota)) \otimes id_Y$$, where $$\iota$$ is the map $$X \otimes X^* \to \mathbb{1}$$ coming from duality data for $$X$$. Here $$Y$$ ranges over all objects in $$\mathcal{C}$$. However, I have no idea if the resulting category would satisfy all the requirements listed above.

EDIT: Inverting $$\Sigma X$$ is clearly equivalent to inverting $$X$$.

In the case where $$\mathcal{C}$$ is presentable, this is constructed in proposition 2.9 of
Robalo, Marco, $$K$$-theory and the bridge from motives to noncommutative motives, Adv. Math. 269, 399-550 (2015). ZBL1315.14030.
Note that if $$\mathcal{C}$$ is stable, so is $$\mathcal{C}[X^{-1}]$$, since $$\mathcal{C}$$ contains an inverse for $$S^1$$ and $$\mathcal{C}\to \mathcal{C}[X^{-1}]$$ is symmetric monoidal.
Furthermore, corollary 2.22 shows that if $$X$$ is a symmetric object (i.e. the cyclic permutation acts trivially on $$X^{\otimes n}$$ for some $$n>1$$), then $$\mathcal{C}[X^{-1}]$$ can be obtained as a category of spectrum objects, i.e. as the $$\infty$$-category $$\operatorname{Stab}_X(\mathcal{C}):=\operatorname{colim}\left(\mathcal{C}\xrightarrow{X\otimes - }\mathcal{C}\xrightarrow{X\otimes-}\cdots\right)\cong \operatorname{lim}\left(\mathcal{C}\xleftarrow{\operatorname{hom}(X,-)}\mathcal{C}\xleftarrow{\operatorname{hom}(X,-)}\cdots\right)$$ where the first colimit is taken in $$\mathrm{Pr}^L$$.
To see an example where the latter identification does not hold, take $$\mathcal{C}=\mathrm{Sp}$$ and $$X=\mathbb{S}\oplus \mathbb{S}$$. Then it can be shown that $$\mathcal{C}[X^{-1}]=0$$ (indeed every additive $$\infty$$-category where the functor $$y\mapsto y\oplus y$$ is fully faithful is 0), but $$\operatorname{Stab}_X\mathcal{C}$$ is not 0, since its K-theory is $$K(\mathbb{S})[1/2]$$. For more discussion of this particular example see this paper.
• If $X$ is not symmetric I assume the categories will not coincide. Will $Stab_X(\mathcal{C})$ be symmetric monoidal and stable when $X$ is not cyclic? – leibnewtz Feb 25 at 7:58