  As a dual of the Auslander transpose of modules, we introduce and study the
cotranspose of modules with respect to a semidualizing module $C$. Then using
it we introduce $n$-$C$-cotorsionfree modules, and show that
$n$-$C$-cotorsionfree modules possess many dual properties of $n$-torsionfree
modules. In particular, we show that $n$-$C$-cotorsionfree modules are useful
in characterizing the Bass class and investigating the approximation theory for
modules. Moreover, we study $n$-cotorsionfree modules over artin algebras and
answer negatively an open question of Huang and Huang posed in 2012.
