For a nice-enough category \(\mathscr {C} \) , we construct both the morphism category \(\textrm{H}(\mathscr {C} )\) of \(\mathscr {C} \) and the category \({{\mathrm{mod{-}}}}\mathscr {C} \) of all finitely presented contravariant additive functors over \(\mathscr {C} \) with values in Abelian groups. The main theme of this paper is to translate some representation-theoretic attributes back and forth between \(\textrm{H}(\mathscr {C} )\) and \({{\mathrm{mod{-}}}}\mathscr {C} \) via the cokernel functor. We consider different exact structures on \(\textrm{H}(\mathscr {C} )\) , and discuss when \(\textrm{H}(\mathscr {C} )\) , endowed with either of these structures, admits almost split sequences, and show that this cokernel functor preserves most of almost split sequences. We apply our results to the case of functorially finite subcategories of module categories to obtain a certain auto-equivalence over them. It turns out that this auto-equivalence is naturally isomorphic to the identity functor when we restrict to functorially finite wide subcategories. This is used to generalize particular types of exact sequences that involve the Nakayama functor and the Auslander-Reiten translation, to the setting of functor categories. Another part of the paper deals with Auslander algebras arising from algebras of finite representation type. In fact, we apply our results to describe the Auslander-Reiten translates of simple modules over Auslander algebras or, in other words, the Auslander-Reiten translate of simple functors. We also describe particular connected components in the Auslander-Reiten quivers of Auslander algebras of self-injective algebras of finite representation type. Further, we state some results on \(\tau \) -periodicity of simple modules over the Auslander algebras arising from self-injective algebras of finite representation type, where \(\tau \) is the Auslander-Reiten translation.
