Say $X = \C$ and $U = \C^*$, $j: U \to \C$. Do you know what is $j^*$ and $j^!$ of $O_X$?

https://arxiv.org/pdf/1607.02064.pdf

This paper discusses general stuff. Very readable, except the notation needs some getting used to. The setup is as following : $X$ is some stable $\infty$-category. No need to assume compactly generated. $U$ is some subcategory which is both reflexive and coreflexive, which means $j_*$ admits both left-adjoint $j^*$ and right-adjoint $j^!$ (I don't know which term corresponds to which condition)

We define $Z^L$ and $Z^R$ as subcategory of $X$. $Z^L = Z^\vee$, is something $\Hom(Z^L, U) = 0$. Similarly, $Z^R = Z^\wedge$ satisfies $\Hom(U, Z^R)=0$.

Consider the original example. Consider $\C[x]/(x) = O_0$. Does $U$ map to it?