  Let $(R, \m)$ be a $d$-dimensional commutative noetherian local ring. Let
$\M$ denote the morphism category of finitely generated $R$-modules and let
$\Sc$ be the submodule category of $\M$. In this paper, we specify the
Auslander transpose in submodule category $\Sc$. It will turn out that the
Auslander transpose in this category can be described explicitly within ${\rm
mod}R$, the category of finitely generated $R$-modules. This result is
exploited to study the linkage theory as well as the Auslander-Reiten theory in
$\Sc$. Indeed, a characterization of horizontally linked morphisms in terms of
module category is given. In addition, motivated by a result of Ringel and
Schmidmeier, we show that the Auslander-Reiten translations in the
subcategories $\HH$ and $\G$, consisting of all morphisms which are maximal
Cohen-Macaulay $R$-modules and Gorenstein projective morphisms, respectively,
may be computed within ${\rm mod}R$ via $\G$-covers. Corresponding result for
subcategory of epimorphisms in $\HH$ is also obtained.
