  Auslander-Reiten duality for module categories is generalized to some
sufficiently nice subcategories. In particular, our consideration works for
$\mathcal{P}^{<\infty}(\Lambda)$, the subcategory consisting of finitely
generated modules with finite projective dimension over an artin algebra
$\Lambda$, and also, the subcategory of Gorenstein projectove modules of
$\rm{mod}\mbox{-}\Lambda$, denoted by $\rm{Gprj}\mbox{-}\Lambda$. In this
paper, we give a method to compute the Auslander-Reiten translation in
$\mathcal{P}^{<\infty}(\Lambda)$ whenever $\Lambda$ is a $1$-Gorenstein
algebra. In addition, we characterize when the Auslander-Reiten translation in
$\rm{Gprj}\mbox{-}\Lambda$ is the first syzygy and provide many algebras having
such property.
