  We show that the category of cancellative conjugation semigroups is weakly
Mal'tsev and give a characterization of all admissible diagrams there. In the
category of cancellative conjugation monoids we describe, for Schreier split
epimorphisms with codomain B and kernel X, all morphisms h from X to B which
induce a reflexive graph, an internal category or an internal groupoid. We
describe Schreier split epimorphisms in terms of external actions and consider
the notions of precrossed semimodule, crossed semimodule and crossed module in
the context of cancellative conjugation monoids. In this category we prove that
a relative version of the so-called "Smith is Huq" condition for Schreier split
epimorphisms holds as well as other relative conditions.
