  Parikh-collinear morphisms have the property that all the Parikh vectors of
the images of letters are collinear, i.e., the associated adjacency matrix has
rank 1. In the conference DLT-WORDS 2023 we showed that fixed points of
Parikh-collinear morphisms are automatic. We also showed that the abelian
complexity function of a binary fixed point of such a morphism is automatic
under some assumptions. In this note, we fully generalize the latter result.
Namely, we show that the abelian complexity function of a fixed point of an
arbitrary, possibly erasing, Parikh-collinear morphism is automatic.
Furthermore, a deterministic finite automaton with output generating this
abelian complexity function is provided by an effective procedure. To that end,
we discuss the constant of recognizability of a morphism and the related
cutting set.
