Dualizability via internal duality

Corollary 11.1.13. Let \(X\) be an object of \(C\) for which the internal hom functor \(\iHom (X,-)\colon C \to C\) exists. Then \(X\) is dualizable if and only if for every \(Z \in C\) the natural map \(Z \otimes D(X) \to \iHom (X,Z)\) adjoint to the composite \(Z \otimes D(X) \otimes X \xrightarrow {Z \otimes \ev } Z \otimes \unit \simeq Z\) is an equivalence.