ct.category theory – Invert a suspension object in a stable monoidal category

Suppose we are given a stable symmetrical full closed monoidal $ ( infty, 1) $-Category $ mathcal {C} $ with suspension $ Sigma $and let $ X in mathcal {C} $ to be dualisable. I would like to create a new stable monoidal symmetric cocomplete category $ mathcal {C}[Sigma X^{-1}]$ at the same time as a symmetrical and symmetric monoidal 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 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} $ who send $ Sigma X $ to an invertible object in $ mathcal {D} $. Here the notation $ Fun_ {ex, cont} ^ { otimes} ( mathcal {C}, mathcal {D}) $ refers to the category of monoidal functors that are both accurate and continuous.

My idea was to locate the card game $ ( Sigma ( iota)) otimes id_Y $, or $ iota $ is the map $ X otimes X ^ * to mathbb {1} $ derived from duality data for $ X $. Right here $ Y $ extends on all objects in $ mathcal {C} $. However, I do not know if the resulting category would meet all the requirements listed above.

EDIT: Invert $ Sigma X $ is clearly equivalent to reverse $ X $.