  The paper concerns a new method to obtain a direct proof of the openness at
linear rate/metric regularity of composite set-valued maps on metric spaces by
the unification and refinement of several methods developed somehow separately
in several works of the authors. In fact, this work is a synthesis and a
precise specialization to a general situation of some techniques explored in
the last years in the literature. In turn, these techniques are based on
several important concepts (like error bounds, lower semicontinuous envelope of
a set-valued map, local composition stability of multifunctions) and allow us
to obtain two new proofs of a recent result having deep roots in the topic of
regularity of mappings. Moreover, we make clear the idea that it is possible to
use (co)derivative conditions as tools of proof for openness results in very
general situations.
