Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor
$$\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$
I don't know much about 2-categories (I'm probably thinking about (2,1)-categories, to be more precise), but I wonder if we may consider a 2-colimit of this and obtain an equivalence of categories
$$\operatorname{2-colim}_U\textsf{QCoh}(U) \xrightarrow{\sim} \textsf{Vect}(K).$$
If this is true, I wonder moreover if the natural functor
$$\textsf{QCoh}(X) \to \operatorname{2-colim}_U\textsf{QCoh}(U)\cong \textsf{Vect}(K)$$
is the one which sends a quasi-coherent module over $X$ to its stalk on the generic point.