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.

翻译：对于从范畴$C$到有序集与保序映射范畴$Pos$的函子$Q$，我们研究将各类结构从基范畴$C$提升到总范畴（或格罗滕迪克范畴）$\int Q$的问题。将幺半结构提升对应于赋予某种使$Q$近乎幺半的松弛自然变换，这或许是范畴论中的常识性结果。我们依赖并推广支撑这一对应关系的工具，从而为提升对称幺半闭结构与星-自主结构提供精确条件。这些刻画的一个推论是：若$Q$通过$SLatt$（完备格与保并映射范畴）分解为幺半函子，则$\int Q$总是对称幺半闭的。在此情形下，我们还提供一种基于量子理论中双重否定核的方法，将$\int Q$转化为星-自主范畴。本文发展的理论最初源于Schalk和de Paiva提出的$P-Set$范畴，它广泛推广了Hyland与Schalk通过正交性结构构造星-自主范畴的方法。