$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$I've seen it written (for example, in Gaitsgory–Rozenblyum) that for a scheme $X$, the category $\IndCoh(X)$ is to be thought of as a categorificaiton of the space of distributions on $X$, just as the (derived) category of quasi-coherent sheaves is to be thought of as a categorification of functions on $X$.

Why exactly is $\IndCoh(X)$ to be thought of as dual to $\QCoh(X)$? The only relation between these categories that I know of is that $\IndCoh(X)$ is a module over $\QCoh(X)$. Is there some way that an ind-coherent sheaf can act on a quasi-coherent one to yield a categorical analog of a point (for example, an object in $\QCoh(\{ \mathrm{pt} \})$)?