nLab
dualizable object

An object $A$ in a monoidal category $C$ is dualizable if it has an adjoint when regarded as a 1-cell in the one-object bicategory $BC$ corresponding to $C$. Its adjoint in $BC$ is called its dual in $C$ and often written as $A^{*}$.

If $C$ is braided then left and right adjoints in $BC$ are equivalent; otherwise one speaks of $A$ being left dualizable or right dualizable. Unfortunately, conventions on left and right vary and sometimes contradict their use for adjoints. But if we define right duals to be the same as right adjoints in $BC$, then a right dual of $A$ is an object $A^{*}$ equipped with a unit (or coevaluation)

$$i:I\to A^{*}\otimes A$$i: I \to A^* \otimes A

and counit (or evaluation)

$$e:A\otimes A^{*}\to I$$e : A \otimes A^* \to I

satisfying the ‘triangle identities’ familiar from the concept of adjunction.

If every object of $C$ has a left and right dual, then $C$ is called a rigid monoidal category or an autonomous monoidal category. If it is additionally symmetric, it is called a compact closed category. See category with duals for more discussion.

Dualizable objects also support a good abstract notion of trace.