EDIT: Note that I have cross-posted the question on MathOverflow in the meantime.
1. Question
Any rigid cartesian monoidal category is trivial (see here). Star-autonomity 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 $\phi_{X,Y,Z}: Hom_C(X \otimes Y,SZ) \xrightarrow{\sim} Hom_C(X, S(Y \otimes Z))$ 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.