1. Question
The n-Lab article on the Chu-construction says:
"Armed with just this much knowledge, and knowledge of how star-autonomous categories behave (as categorified versions of Boolean algebras, or perhaps better Boolean rigs), the star-autonomous structure on $Chu(C,d)$ can pretty much be deduced (or strongly guessed) […]."
How do star-autonomous categories behave as categorified versions of Boolean algebras or Boolean rigs?
2. Wikipedia says
One explanation might be given on wikipedia:
"A degenerate example [of a star-autonomous category] (all homsets of cardinality at most one) is given by any Boolean algebra (as a partially ordered set) made monoidal using conjunction for the tensor product and taking 0 as the dualizing object."
I suppose the internal hom of two objects $a,b$ in this category is $\neg a \lor b $, correct? The dual functor is the complement?