  Almost split sequences lie in the heart of Auslander-Reiten theory. This
paper deals with the structure of almost split sequences with certain ending
terms in the morphism category of an Artin algebra $\Lambda$. Firstly we try to
interpret the Auslander-Reiten translates of particular objects in the morphism
category in terms of the Auslander-Reiten translations within the category of
$\Lambda$-modules, and then use them to calculate almost split sequences. In
classical representation theory of algebras, it is quite important to recognize
the midterms of almost split sequences. As such, another part of the paper is
devoted to discuss the midterm of certain almost split sequences in the
morphism category of $\Lambda$. As an application, we restrict in the last part
of the paper to self-injective algebras and present a structural theorem that
illuminates a link between representation-finite morphism categories and Dynkin
diagrams.
