Dualizable object

Definition 11.1.1 (Dualizable object). Consider objects \(X\) and \(Y\) of \(C\). We say that a morphism \(\ev \colon Y \otimes X \to \unit \) exhibits \(Y\) as a dual to \(X\) if there exists another morphism \(\coev \colon \unit \to X \otimes Y\) such that the triangle identities are satisfied: there exist commutative diagrams

In this case we refer to \(\ev \) as the evaluation map and to \(\coev \) as the coevaluation map, and call \((Y,\ev )\) a duality datum for \(X\). We say that an object \(X\) is dualizable if it admits such a duality datum.