  In this article, we study the ideals of mid $p$-summing operators. We obtain
representation of these operator ideals by tensor norms. These tensor norms are
defined by using a particular kind of sequential dual of the class of mid
$p$-summable sequences. As a consequence, we prove a characterization of the
adjoints of weakly and absolutely mid $p$-summing operators in terms of the
operators that are defined by the transformation of dual spaces of certain
vector-valued sequence spaces.
