  Let $G$ be a finite group, and $C$ be an abelian group. We introduce the
notions of $C$-monomial $G$-sets and $C$-monomial $G$-posets, and state some of
their categorical properties. This gives in particular a new description of the
$C$-monomial Burnside ring $B_C(G)$. We also introduce Lefschetz invariants of
$C$-monomial $G$-posets, which are elements of $B_C(G)$. These invariants allow
for a definition of a generalized tensor induction multiplicative map
$\mathcal{T}_{U,\lambda}: B_C(G)\to B_C(H)$ associated to any $C$-monomial
$(G,H)$-biset $(U,\lambda)$, which in turn gives a group homomorphism
$B_C(G)^\times\to B_C(H)^\times$ between the unit groups of $C$-monomial
Burnside rings.
