Currying and uncurrying functors

Remark 1.3.18. Given a functor \(g\colon E \to \Fun (C,D)\), we define its uncurrying \(g^u\colon E \times C \to D\) as the composite \[ E \times C \xrightarrow {g \times \id _C} \Fun (C,D) \times C \xrightarrow {\ev } D. \] Thus currying and uncurrying pass between functors \(E \times C \to D\) and functors \(E \to \Fun (C,D)\), and going back and forth both ways is naturally isomorphic to the identity. In the case \(E = *\), we observe that objects \(* \to \Fun (C, D)\) of the functor category correspond to functors \(C \simeq C \times * \to D\) from \(C\) to \(D\), justifying the terminology for \(\Fun (C,D)\).