  Let $\mathcal{F}$ be a coherent sheaf on a complex variety $X$ that has a
locally free resolution $E^{\bullet}$. In [19], the authors constructed a
pseudomeromorphic current whose support is contained in $supp(E^{\bullet})$
that represents products of Chern classes of $\mathcal{F}.$ In this paper, we
show that their construction works for general de-Rham characteristic classes
and then generalize it to represent products (in de-Rham cohomology) of
characteristic forms of cohesive modules defined by Block. Finally, we state a
corollary to a transgression result in [16] that show that it is sufficient to
only use the degree-$0$ and degree-$1$ parts of the superconnection to
construct currents that represent characteristic forms of cohesive modules in
the Bott-Chern cohomology.
