  Categorical constructions on heaps and modules over trusses are considered
and contrasted with the corresponding constructions on groups and rings. These
include explicit description of free heaps and free Abelian heaps, coproducts
or direct sums of Abelian heaps and modules over trusses, and description and
analysis of free modules over trusses. It is shown that the direct sum of two
non-empty Abelian heaps is always infinite and isomorphic to the heap
associated to the direct sums of the group retracts of both heaps and
$\mathbb{Z}$. Direct sum is used to extend a given truss to a ring-type truss
or a unital truss (or both). Free modules are constructed as direct sums of a
truss. It is shown that only free rank-one modules are free as modules over the
associated truss. On the other hand, if a (finitely generated) module over a
truss associated to a ring is free, then so is the corresponding
quotient-by-absorbers module over this ring.
