Given a scheme $X$  and divisor $D\subseteq X$, you can take the line bundle
$$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \mathcal{M}$$
where $\mathcal{M}$ is the sheaf of rational functions on $X$. It satisfies $\mathcal{O}_X(D+D') =\mathcal{O}_X(D)\otimes\mathcal{O}_X(D')$. Moreover, under weak conditions these are the only coherent sheaves with an inverse under $\otimes$. 

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One categorical level up, you can consider sheaves of *categories* on $X$, with $\text{QCoh}_X$ being the analogue of $\mathcal{O}_X$.

My **question** is: what is the analogue "$\text{QCoh}_X(D)$" of $\mathcal{O}_X(D)$? 

Ideally, the definition will mimic the above, with $\text{QCoh}_X(D)$ being a subsheaf of categories of $j_*\text{QCoh}_\eta$ defined by similar [conditions], and having $\text{QCoh}_X(D)\otimes\text{QCoh}(D')=\text{QCoh}(D+D')$. Ideally, under the same weak conditions as above, these are the only invertible sheaves of categories.