  Let M, N be monoids, and PSh(M), PSh(N) their respective categories of right
actions on sets. In this paper, we systematically investigate correspondences
between properties of geometric morphisms PSh(M) $\rightarrow$ PSh(N) and
properties of the semigroup homomorphisms M $\rightarrow$ N or
flat-left-N-right-M-sets inducing them. More specifically, we consider
properties of geometric morphisms featuring in factorization systems, namely:
surjections, inclusions, localic morphisms, hyperconnected morphisms,
terminal-connected morphisms, {\'e}tale morphisms, pure morphisms and complete
spreads. We end with an application to topos-theoretic Galois theory to the
special case of toposes of the form PSh(M).
