  Connections between heaps of modules and (affine) modules over rings are
explored. This leads to explicit, often constructive, descriptions of some
categorical constructions and properties that are implicit in universal algebra
and algebraic theories. In particular, it is shown that the category of groups
with a compatible action of a truss T (also called pointed T-modules) is
isomorphic to the category of modules over the ring R(T) universally associated
to the truss. This is widely used in the explicit description of free objects.
Next, it is proven that the category of heaps of modules over T is isomorphic
to the category of affine modules over R(T) and, in order to make the picture
complete, that (in the unital case) these are in turn equivalent to a specific
subcategory of the slice category of pointed T-modules over R(T). These
correspondences and properties are then used to describe explicitly various
(co)limits and to compare short exact sequences in the Barr-exact category of
heaps of T-modules with short exact sequences as defined previously.
