  We investigate the structure of certain almost split sequences in
$\mathcal{P}(\Lambda)$, i.e., the category of morphisms between projective
modules over an Artin algebra $\Lambda$. The category $\mathcal{P}(\Lambda)$
has very nice properties and is closely related to $\tau$-tilting theory,
$g$-vectors, and Auslander-Reiten theory. We provide explicit constructions of
certain almost split sequences ending at or starting from certain objects.
Applications, such as to $g$-vectors, are given. As a byproduct, we also show
that there exists an injection from Morita equivalence classes of Artin
algebras to equivalence classes of 0-Auslander exact categories.
