# star-autonomous closed categories

A $\ast$-autonomous category is usually defined 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 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, 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).

Barr's "Definition D" in non-symmetric $\ast$-autonomous categories 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 $[A,[B^*,C^*]] \cong [[A,B]^*,C^*]$.