Exact module categories over $\mathrm{Rep}(u_q(\mathfrak{sl}_2))$

We give a complete list of indecomposable exact module categories over the finite tensor category $\mathrm{Rep}(u_q(\mathfrak{sl}_2))$ of representations of the small quantum group $u_q(\mathfrak{sl}_2)$, where $q$ is a root of unity of odd order. Each of them is given as the category of representations of a left comodule algebra over $u_q(\mathfrak{sl}_2)$ explicitly presented by generators and relations.