  For a functor $Q$ from a category $C$ to the category $Pos$ of ordered sets
and order-preserving functions, we study liftings of various kind of structures
from the base category $C$ to the total(or Grothendieck) category $\int Q$.
That lifting a monoidal structure corresponds to giving some lax natural
transformation making $Q$ almost monoidal, might be part of folklore in
category theory.We rely on and generalize the tools supporting this
correspondence so to provide exact conditions for lifting symmetric monoidal
closed and star-autonomous structures.A corollary of these characterizations is
that, if $Q$ factors as a monoidal functor through $SLatt$, the category of
complete lattices and sup-preserving functions, then $\int Q$ is always
symmetric monoidalclosed. In this case, we also provide a method, based on the
double negation nucleus from quantale theory, to turn $\int Q$ into a
star-autonomous category.The theory developed, originally motivated from the
categories $P-Set$ of Schalk and de Paiva, yields a wide generalization of
Hyland and Schalk construction of star-autonomous categories by means of
orthogonality structures.
