Internal hom object

Definition 11.1.8. Given two objects \(X,Z \in C\), an object \(\iHom (X,Z) \in C\) equipped with a map \(\ev \colon \iHom (X,Z) \otimes X \to Z\) is said to exhibit \(\iHom (X,Z)\) as an internal hom from \(X\) to \(Z\) if for every third object \(W\) the composite \[ \Hom _{C}(W,\iHom (X,Z)) \xrightarrow {- \otimes X} \Hom _{C}(W \otimes X, \iHom (X,Z) \otimes X) \xrightarrow {\ev \circ -} \Hom _{C}(W \otimes X, Z) \] is an equivalence.