  In the context of protomodular categories, several additional conditions have
been considered in order to obtain a closer group-like behavior. Among them are
locally algebraic cartesian closedness and algebraic coherence. The recent
notion of S-protomodular category, whose main examples are the category of
monoids and, more generally, categories of monoids with operations and
Jo\'{o}nsson-Tarski varieties, raises a similar question: how to get a
description of S-protomodular categories with a strong monoid-like behavior. In
this paper we consider relative versions of the conditions mentioned above, in
order to exhibit the parallelism with the "absolute" protomodular context and
to obtain a hierarchy among S-protomodular categories.
