Disclaimer: This is a crosspost (see MathStackexchange). Apologies if cross-posting is frowned upon. However, it seems that on Stackexchange there are not many people familiar with star-autonomous categories.

**1. Question**

Any rigid cartesian monoidal category is trivial (see here). Star-autonomy is a generalization of rigidity. Are there (non-trivial) examples of star-autonomous cartesian monoidal categories? Neither Zhen Lin's nor Martin Brandenburg's arguments for the rigid case seem to be easily adaptable to star-autonomous categories.

**2. Additional remarks**

I use the following definition:

A monoidal category $(C, \otimes,I)$ carries a *star-autonomous structure* if there exists an equivalence of categories $S: C^{op} \xrightarrow{\sim} C$ with inverse $S’$ such that there there are bijections $\phi_{X,Y,Z}: \operatorname{Hom}_C(X \otimes Y,SZ) \xrightarrow{\sim} \operatorname{Hom}_C(X, S(Y \otimes Z))$ natural in $X,Y,Z.$

I tried the category **Pos** of posets and monotone maps. This category is cartesian closed. The product is the product order. The set of monotone maps between two posets becomes a poset by setting $$f \leq g \text{ for } f,g: X \rightarrow Y \text{ if } f(x)\leq g(x) \text{ for all } x\in X.$$ Does this category admit a star-autonomous structure? Is it even self-dual? If on objects one sets $S(X):=X^{op}$ with $X^{op}$ the dual poset, it is not clear to me how to define the mapping on morphisms in order to obtain a duality functor. In the category of sup-lattices such a definition is possible, but it uses the existence of joins.