A $\ast$-autonomous category is [usually defined](https://ncatlab.org/nlab/show/star-autonomous+category) as a *monoidal* category with some extra structure, including that it is closed monoidal (depending on the definition this may be assumed or proved). However, in many examples (such as sup-lattices) it seems that the closed structure is more fundamental, with the tensor product being defined from this as $X\otimes Y = [X,Y^\*]^\*$.


Is there a notion of $\ast$-autonomous [closed category](https://ncatlab.org/nlab/show/closed+category) that is equivalent to the usual notion of $\ast$-autonomous category, but where the axioms are expressed only in terms of the closed structure? It might involve a [dualizing object in a closed category](https://ncatlab.org/nlab/show/dualizing+object+in+a+closed+category), but offhand that doesn't seem quite enough; also my own inclination would be to axiomatize the operation $(-)^\*$ rather than the dualizing object (since this also is what I see more naturally in examples like suplattices).



star-autonomous closed categories
Barr's "Definition D" in [non-symmetric $\ast$-autonomous categories](http://www.math.mcgill.ca/barr/papers/asymm.pdf) is a closed category $\mathcal{A}$ together with an $\mathcal{A}$-enriched equivalence $(-)^\* : \mathcal{A}^{\mathrm{op}} \simeq \mathcal{A}$ and an $\mathcal{A}$-enriched natural isomorphism $[A,[B^\*,C^\*]] \cong [[A,B]^\*,C^\*]$.


