A weak $*$-autonomous category is a symmetricclosed monoidal category$\langle
C,\otimes, I,\multimap\rangle$ with an object $\bot$ such that the canonical morphism

A $*$-autonomous category is a weak $*$-autonomous category such that for every object $A$, $d_{A}: A \to (A \multimap \bot) \multimap \bot$ is an isomorphism.

A strictly weak $*$-autonomous category is a weak $*$-autonomous category such that for at least one object $A$, $d_{A}$ is not an isomorphism. Equivalently, a strictly weak $*$-autonomous category is a weak $*$-autonomous category which is not a $*$-autonomous category.

Example

For every field$\mathbb{K}$, the category $Vec_{\mathbb{K}}$ is a strict weak $*$-autonomous category.

For every commutative ring$R$, the category $Mod_{R}$ is a strict weak $*$-autonomous category.