  Let $\mathsf{Rep}(H)$ be the category of finite-dimensional representations
of a finite-dimensional Hopf algebra $H$. Andruskiewitsch and Mombelli proved
in 2007 that each indecomposable exact $\mathsf{Rep}(H)$-module category has
form $\mathsf{Rep}(B)$ for some indecomposable exact left $H$-comodule algebra
$B$. This paper reconstructs and determines a quasi-Hopf algebra structure from
the dual tensor category of $\mathsf{Rep}(H)$ with respect to
$\mathsf{Rep}(B)$, when $B$ is a left coideal subalgebra of $H$. Consequently,
it is categorically Morita equivalent to $H$, and some other elementary
properties are also studied. As applications, our construction could be applied
to imply some classical results on bismash products of matched pair of groups
and bosonizations of dually paired Hopf algebras.
